The person I heard the last puzzle (P122636) from attributed it to https://x.com/cshearer41 although I haven't found the tweet to confirm. Here's one of her most popular ones.

Question: What's the total shaded area?

https://teorth.github.io/equational_theories/blueprint.pdf

What do you think about this?

Let's try a new problem!
Three squares are arranged as shown in the diagram. The green square has an area of 12. What's the area of the yellow square?

I like playing around with formal logic, but when proving things about synthetic geometry in the usual kinds of formal logic, you sometimes have to spend time proving things that would be obvious from a diagram. It's often stated that diagrams can't be part of a formal proof because they can deceive you into thinking you've proved false things, such as the infamous proof that all triangles are isosceles. But in all the false proofs using diagrams I've seen, the problem was that the diagram didn't consider all possible cases. There's no reason a system of logic using diagrams couldn't be made just as sound as regular predicate logic.

So I've started reading PDF related, which describes such a system of diagrams for proving things about geometry. It's a short book made from an edited version of his PhD thesis. Apparently people have also been studying systems of diagrammatic reasoning not specific to geometry such as Venn diagrams and similar systems, including proving various soundness results about them. One criticism I've seen about the system in the book is that the number of cases you have to draw can be very large, and you can see one instance of that in the book in Figure 9. But at least the operation that leads to Figure 9 is not something you'd normally do in a proof, so I don't know yet how much of a problem case explosion is in practice. I'm hopeful that this system can and will be improved upon, and that in general diagrams will become a more common tool in formal proofs.

Please advise the literature for studying the theory of formal languages, especially the theory of automata (preferably in Russian)

The bunkbed conjecture is false https://arxiv.org/abs/2410.02545
How can a path being connected up to the last post not be independent of the 2 possible paths being connected following it?

Pic related is a typical high school geometry exercise to discover the sum of the interior angles of a polygon. To their credit, the authors only claim that what they have shown works for convex polygons. It is left to the reader to wonder if the trick can be adapted to polygons in general.

Without looking up the answer, can you show with proof whether any n-sided polygon can be decomposed into n-2 triangles?

It has been proven that the longest-running 5-state 2-symbol Turing machine which halts is the one that halts in 47,176,870 steps. All 5-state 2-symbol Turing machines which run longer than this have been proven to run forever.

The halting theorem famously proves that there's no general algorithm that can determine whether a given computer program halts. Some programs we know halt because we can run them to completion. Others enter obvious infinite loops. But for some programs, it's hard to tell if they eventually halt or just run for a very long time. It's tantalizing to wonder where the boundary point is between simple programs whose halting status we can figure out and those programs we can't.

Research into the busy beaver problem probes and pushes that boundary. When I posted about it a year ago in P52827, there were several remaining 5-symbol 2-state Turing machines whose halting status was unknown. They were conjectured to run forever, but it was not proven. Now it's been proven, and the frontier has been pushed out to BB(6).

Homepage of the collaboration that proved it:
https://bbchallenge.org/
Announcement of the discovery:
https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-proved-bb-5-47-176-870/237
Machine-verifiable formal proof of BB(5) = 47,176,870 in Coq:
https://github.com/ccz181078/Coq-BB5
A well-written news story on the feat (sadly behind Cloudflare):
https://www.quantamagazine.org/amateur-mathematicians-find-fifth-busy-beaver-turing-machine-20240702/

Let A be finite.
Consider a uniformly distributed random function f: A->A.
How does [tex: N(n) := E[|f^n(A)|]] vary with n and |A|?

Is lambda smart enough for Lean's Natural Numbers Game? Or anyone here already completed it (it's not too hard tbh)?

https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_game/index2.html

Basically, you program in Lean, a programming "language"/proof assistant used by autists and mathematicians. Your goal is to prove some profound shit like "a + b = b + a", starting from nothing except peano axioms. It's more fun than it sounds like. And is a good introduction to formal logic.

Yes, it works in TB but it's *****ing slow to load (some 20MB of library stuff to download). And you need to allow canvas access for the main menu to work.

PS: there's also a new version updated to Lean 4 but it's cringe and dumbed down and gay and doesn't even work with tor. Trust me, ignore it and just try this one.

>A vector space just generalizes a field.
<No? Vectors are a whole separate set of objects you introduce. It's not just a generalization.
>But if you make the vector space the same field as the scalars it's the same.
<But vectors are different from scalars.
>But it's the same object.
<But they're different types. We write vectors in bold.
>Math doesn't have types.

>The next simplest example is the field F itself. Vector addition is just field addition, and scalar multiplication is just field multiplication. This property can be used to prove that a field is a vector space. Any non-zero element of F serves as a basis so F is a 1-dimensional vector space over itself.
...
Surely it causes problems if you can't assume vectors and scalars are distinct?

This is everything I hate about mathematics in one example. Are mathematicians not capable even of using tagged unions? Imagine if the vector space contained the objects used to construct the scalar field. It's insane and will make mathematics implode under special cases eventually. It's like math isn't made to find mathematical truths directly but so some "wiggers" (jewish) can go "UM ACKCHUALLY" and use intentionally hidden structure to make some important seeming abstractions in a way that somehow avoids helping anyone else understand. It's utterly disgusting.
Post other examples of this stuff itt, if you want.

>use the [quadratic] formula as a last resort
/watch?v=ij01aenZIfY

The video is a typical example of this bad advice. She tells her students to try to use integer factor searching to solve [tex: 3x^2 - 4x - 4 = 0], [tex: 5x^2 - 2x + 3 = 0], and [tex: 2x^2 - 3x - 7 = 0] before proceeding to the quadratic formula. Even with her contrived examples (the factor search works for one of them) you can see she's wasting time. But the harm isn't the wasted time; it's that some of her students will eventually see an equation like [tex: x^2 - 5.6x - 3.784 = 0], think that they're supposed to solve it by factoring, and declare "I don't know how to do this" rather than moving on to the quadratic formula as she intended.

i never understood all the buzz around the barber paradox
>the barber shaves all those and only those who do not shave themselves, and everybody must be shaved
obviously the barber shaves because thats his job as the barber
however, when he gets home, he is no longer performing his job
and he cannot go to his shop bc by the time his shift ends, the barbershop is closed
so he shaves himself like the other men who do not go to the barbershop
wheres the paradox? are mathematicians stupid or just extremely *****ing autistic?

It has been a long time since we've had one of these, so here is an easy puzzle:
How can you arrange 12 squares to form a net for a single cube? You may not cut them, superpose them, or leave any unused.

how do you factor something like 6x^2 + 12 x + 6 (from (2x + 2) * (3x + 3))
every time i learn this i forget it :( too lazy to open my textbook rn

If the title made you think if of having a stroke, this thread is to discuss two of the problems presented in this year's high-school exit exams. In specific, for higher level mathematics (the proper one, not social science mathematics). So you should be able to solve these. Also it's worth mentioning that the problems have been slightly refitted. Rest assured, the contents are exactly the same.

Problem 1:

Take [tex: f(x) = (1-ax)^{\frac{-1}{2}}], where ax<0, [tex: a \neq 0]

a, Prove using induction that
[tex: f^{(n)}(x) = \frac{a^n(2n-1)!(1-ax)^{\frac{2n+1}{2}}}{2^{2n-1}(n-1)!}], [tex: n \in \mathbb{Z}^+]

b, Show that the Maclaurin series (not the Taylor series) for f(x) up to an including the [tex: x^2] term is
[tex: 1 + \frac{1}{2}ax + \frac{3}{8}a^2x^2]

c, Show that
[tex: (1-2x)^{-\frac{1}{2}}(1-4x)^{-\frac{1}{2}} \approx \frac{2+6x+19x^2}{2}]

d, Knowing that the expansion of the Maclaurin series for f(x) is convergent for |ax|<1, indicate what restriction must be put on x for the previous approximation to be valid.

e, use [tex: x=\frac{1}{10}] to make an approximation for the value of [tex: \sqrt x] expressed in terms of [tex: \frac{c}{d}] where [tex: c,d \in \mathbb{Z}^+]

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

Problem 2:

Consider the following ODE:

[tex: \frac{dy}{dx} - y cosec(2x) = \sqrt(tan x)]

a, Use Euler's method with a step size of [tex: \frac{\pi}{12}] to make an approximation for the value of y when [tex: x = \frac{5\pi}{12}]. For added spice, you may not use your calculator here.

b, Show that
[tex: \frac{d}{dx}(\frac{1}{2}ln(cotg x)) = -cosec 2x]

c, Show that [tex: \sqrt(cotg x)] is an integrating factor for this ODE

d, Solve the ODE for y

e, Consider the results from (d) and (a), and that (d) is valid for [tex: 0 < x < \frac{\pi}{2}].
(i) Calculate the exact value of y for [tex: x = \frac{5\pi}{12}]
(ii) Considering the slope of the curve, indicate why (a) is not a good approximation
(iii) Write the reason why the approximation for y is lower than the actual value

f, Using the ODE, deduce why the solution has a positive slope for [tex: 0 < x < \frac{\pi}{2}]

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

I think both are fun to do. Problem 1 focuses on a practical application for a primitive tool for approximation of square roots. Problem 2 is exploring an ODE and testing the student's knowledge on when can approximations be applied.

I've been playing a bit with ternary logic and arithmetic. The initial reason I started thinking about this is that it annoyed me how in binary a n-bit signed integer can represent the range [- 2^(n-1) ; 2^(n-1) - 1], which isn't symmetrical. Of course that's only a convention (although one that is very convenient for several reasons) and it could as well represent [- 2^(n-1) + 1/2 ; 2^(n-1) - 1/2], but then 0 wouldn't be represented.

In ternary, a n-"tit" signed integer can represent all the integers in [-(3^(n-1) - 1)/2 ; (3^(n-1) - 1)/2]. So a 3-tit signed integer can represent the 27 numbers between -13 and +13, for example.

Let's see how to construct such a numbering system. We could use 3's complement for negative numbers. For example, with 2-tit numbers:
12 = -4
20 = -3
21 = -2
22 = -1
00 = 0
01 = 1
02 = 2
10 = 3
11 = 4
Now that's very ugly, and there is no obvious way to determine whether a number is negative or positive from it's ternary representation. I don't like it.

So instead of assigning to each tit a value in {0, 1, 2}, we can instead assign them a value in {-1, 0, 1}. For convenience and legibility, I will note the different symbols as {1̃, 0, 1}, but remember that 1̃ has a value of -1. With those values, and a classic positional system, we can easily represent all natural numbers.
The positive numbers are:
0, 1, 11̃, 10, 11, 11̃1̃, 11̃0, ...
(0, 1, 2, 3, 4, 5, 6, ...)
The negative numbers are:
1̃, 1̃1, 1̃0, 1̃1̃, 1̃11, 1̃10, 1̃11̃, ...
(-1, -2, -3, -4, -5, -6, -7, ...)

This system has some nice properties:
1. All natural numbers have a unique representation
2. The range that can be represented by an n-tit number is symmetrical with regards to 0
3. The leftmost non-0 tit contains the sign information
4. The opposite of a number can be easily computed by replacing all 1s by 1̃s and all 1̃s by 1s
5. It's a positional system, so usual algorithms for addition and multiplication apply.
6. A n-tit variable can represent ~60% more values than a n-bit variable

We can also have a gray-like code. Here's how to generate a sequence where each consecutive term only differs by one tit:
1. Start with the sequence S ← {1̃, 0 1}
2. T and U both take the elements of S
3. S is reversed
4. Each element t of T has the tit 1̃ appended to its left
5. Each element s of S has the tit 0 appended to its left
6. Each element u of U has the tit 1 appended to its left
7. S ← T | S | U where "|" represents the concatenation operator.
8. Redo steps 2 to 7 until the sequence is long enough.

For example, with 2-tit numbers, we get:
1̃1̃ 1̃0 1̃1 01 00 01̃ 11̃ 10 11


Now let's do some logic!
We could easily emulate binary logic, for example by interpreting 0 as a boolean 0 and 1̃ and 1 as a boolean 1. That's not fun though, and wouldn't allow us to compute on ternary numbers, so let's invent toolean logic.
I tried coming up with a few logic gates, as extensions to the ones we're used to with booleans.

"and", it's a ternary multiplication. Notice how the lower-right quadrant is the same as with the boolean and. I'll note the operator as ⋅
 a⋅b | 1̃ 0 1
----+-------
1̃ | 1 0 1̃
0 | 0 0 0
1 | 1̃ 0 1
That one has some nice properties:
(a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) = a ⋅ b ⋅ c
a ⋅ a = abs(a) (that is, 1̃→1, 0→0 and 1→1)
1̃ ⋅ a = neg(a) (that is, 1̃→1, 0→0 and 1→1̃)
1 ⋅ a = a
0 ⋅ a = 0
a ⋅ b = neg(a) ⋅ neg(b)

"or", it's a ternary saturating addition. Again the lower-right quadrant is the same as with the boolean or. I'll note the operator as +
a+b | 1̃ 0 1
----+-------
1̃ | 1̃ 1̃ 0
0 | 1̃ 0 1
1 | 0 1 1
That one isn't as nice, so I would be happy if you have better suggestions. In particular, I hate that (a + b) + c ≠ a + (b + c). Still, a few interesting properties:
neg(a) + neg(b) = neg(a + b)
a + a = a
0 + a = a
a + neg(a) = 0

"xor", isn't an overflowing addition (half adder), because I wanted the lower-right cadrant to stay the same as the boolean xor. I'll note the operator as ⊕
 a⊕b | 1̃ 0 1
----+-------
1̃ | 0 1̃ 0
0 | 1̃ 0 1
1 | 0 1 0
Again, I hate that (a ⊕ b) ⊕ c ≠ a ⊕ (b ⊕ c)
Some properties:
a ⊕ b = (a + b) + ((a + b) ⋅ a ⋅ b) (can this expression be simplified?)
a ⊕ a = 0
0 ⊕ a = a
a ⊕ neg(a) = 0

There is no "not", because why would not(0) be 1 over 1̃? In any case both variant can be implemented as (a ⊕ 1) and (a ⊕ 1̃) respectively.


Can the logic operators do arithmetic? Yes they can!
Here's the truth table for a half adder
| 1̃ 0 1
----+-------
1̃ | 1 1̃ 0
0 | 1̃ 0 1
1 | 0 1 1̃
That is, (a ⊕ b) + ((a + b) ⋅ neg(a ⋅ b)). Can this expression be simplified?

And here's the truth table for a full adder
| 1̃1̃ 1̃0 1̃1 01 00 01̃ 11̃ 10 11
----+------------------------------
1̃ | 1̃0 1̃1 01̃ 00 01̃ 1̃1 01̃ 00 01
0 | 1̃1 01̃ 00 01 00 01̃ 00 01 11̃
1 | 01 00 01 11̃ 01 00 01 11̃ 10
Now I haven't got a clue as to where to start to get a toolean expression out of this.


What do you think about all this? Did I make any mistake?
Some more things I want to look into:
1. Other properties of toolean logic, especially to help find and simplify expressions
2. Find an expression for the full adder
3. Design ternary CMOS circuits
4. Check out what people smarter than me have written, but I want to play around a bit more before.

henlo frens
wat r sum god places 2 git sick epic classic math books 4 cheap?
asking 4 fren of fren

Can someone explain where the following mathy things are used?
I'm interested in where it is used mainly, and also how it's used.

>Fourier Series
I know it's used to solve heat conduction equation and that's why the autist Fourier came up with all this but I'm sure there's more uses to this
>Partial Differential Equations
>Analytic Functions
>Complex Interpretations
>Vector Calculus

Cute fact I learned recently: Start with a triangle ABC and extend sides AB, BC, and AC out to points D, E, and F, respectively. Pick an arbitrary point M on side AB. Draw a line from M to E, and let M' be where it crosses side AC. Continuing, use a line through M' and D to make a point M'' on side BC, then use a line through M'' and F to make a point M''' on side AB. After two circuits around the triangle in this manner, we always come back to the original starting point M.

This is a consequence of Pappus's hexagon theorem. Can you see why?

Have you been studying anything interesting lately? For myself, I've been trying to learn a bit more group theory.

I was wondering if anyone would be down to figure out π and φ again, just because we can and I'm schizo enough to not trust the fact that π is 3.14... and φ is 1.61...

[bold:The π question]
we know that π is used in the calculation of the lenght of the circle:
L=2*π*R
L=D*π

L=Lenght of the Circle
R=Radius of the Circle
D=Diameter of the Circle

In the Bible (KJV here), at verse 1 Kings 7:23 it is said that:
"And he made a molten sea, [bold:ten] cubits from the one brim to the other: it was [bold:round all about], and his height was five cubits: and a line of [bold:thirty] cubits did compass it round about."

it is also repeated in 2 Chronicles 4:2:
"Also he made a molten sea of [bold:ten] cubits from brim to brim, [bold:round in compass], and five cubits the height thereof; and a line of [bold:thirty] cubits did compass it round about."

giving us the values of:
D=10 cubits => R=5 cubits
L=30 cubits
L=D*π <=> 30=10*π => π=3
[bold:Note]:The lenght of the circle is 30 with the diameter being 10; if π was truly the 3.14... value it is given today, then either the value in bible is rounded down from 31.4/31.5 to 30 or the currently recognised value of π is false

Some mathematicians throughout their life attempted to calculate π via polygons, by using the Radius connected from the centre of the Polygon to one of its corners or perpendicular to one of its sides, and the higher the Polygon side count is, the closer to reality the value comes, in theory

The only usable solution to calculating π would be through calculations as attempting IRL experiments would result in false values due to too many physical factors that would throw us off, even slightly


[bold:The φ question]
We know that φ is the value of the golden ratio following this ecuation:
A/B=(A+B)/A as long as A>B>0
A=Lenght of a rectangle
B=Width/Height of a rectangle

The value of φ fluctuated between different values, with it being today at the value of 1.61...

Many mathematicians also figured out that the Pentagon and the Pentagram has φ all over it, so those would be some sources of inspiration as well, I suppose


I'll come from time to time over here to share and give my opinion, as well as the results of my calculations on the size of π & φ, have a wonderful day and good luck at your calculations, cheers! ^^

Someone on 4/sci/ just made a video attempting to explain Uncle Ted's PhD thesis at the level of "a general audience with some calculus familiarity." Did he succeed? How much of it do you understand?

/watch?v=wD4xrnzKN1Y

As part of studying group theory I've been playing around with various concrete groups, among them the rotational symmetry groups of the regular polyhedra. And I noticed something cool, although I'm far from the first to notice it.

In a regular dodecahedron, you can connect certain vertices together to form an inscribed cube. There are five different ways of doing this. If you draw out the edges of the five cubes, each cube in a different color, then on each face of the dodecahedron you end up drawing a pentagram with each of the five line segments in a different color. And on each face of the dodecahedron the colors in the pentagram are arranged differently; in other words, each face displays a different circular permutation of the five colors. There are twenty-four possible circular permutations of five objects, and the twelve permutations seen on the dodecahedron are specifically the even permutations, permutations you can reach with an even number of swaps.

This is an easy way to see that the rotational symmetries of the dodecahedron are isomorphic to the even permutations of five objects. Each rotation in the group of symmetries permutes the five colored cubes, and given an even permutation of the five colors, you can figure out the rotation it corresponds to by picking a face, identifying which face will need to rotate to that face's location to get the desired permutation of colors up to rotation, then seeing how that face will need to be rotated about its center to make the permutation match exactly.

Files related are a picture of the cubes and a foldable version from
https://www.chiark.greenend.org.uk/~sgtatham/polypics/dodec-cubes.html

Given a triangle with one obtuse angle, is it possible to cut the triangle into smaller triangles, all of them acute? (An acute triangle is a triangle with three acute angles. A right angle is of course neither acute nor obtuse.) If this cannot be done, give a proof of impossibility. If it can be done, what is the smallest number of acute triangles into which any obtuse triangle can be dissected?

The illustration shows a typical attempt that leads nowhere. The triangle has been divided into three acute triangles, but the fourth is obtuse, so nothing has been gained by the preceding cuts.

Everyone knows how to prove the pigeonhole principle by induction. Can you prove induction from the pigeonhole principle?

You have a rectangular chocolate bar marked into m x n squares, and you wish to break up the bar into its constituent squares. At each step, you may pick up one piece and break it along any of its marked vertical or horizontal lines.

Prove that every method finishes in the same number of steps.

when you try to add 2 large binary numbers on the computer, they overflow and the computers returns a negative number
when you try to add all natural numbers, you get -1/12
proof we live in a computer simulation?

So apparently Google used AI to discover new ways to troll people in an 8-dimensional Set game, among other mathematical discoveries:

https://github.com/google-deepmind/funsearch

Is this a big step forward in AI or is it just a speedup on problems where they could have used a search algorithm not based on language models to discover similar results? How do you think generative AI will influence mathematics in the coming years?

how do i get god at math
i,m get filtered just trying to prove basic properties of coomplex arithmeitc-

You should be able to solve this.

OH NO!!!!
Autistic Madoka has trapped Rena in the center of a perfectly circular pond. Autistic Madoka is waiting on the perimeter of the pond and can run exactly 4x faster than Rena can swim, and will always move to the part of the shore that is closest to Rena as possible

if Rena can touch land for even a split second before Autistic Madoka catches her she can escape, but if Madoka is waiting for her when she arrives then she'll kill Rena!

HOW TF CAN RENA ESCAPE TO SAVE US???

note: if Autistic Madoka can take two paths of the same distance to the location she'll flip a coin and choose one

Autistic madoka can change directions instantly!

Suppose the cost of shipping a rectangular box is proportional to the sum of its length, width, and height. Is it possible to save money by putting the box you want to deliver inside another box that costs less to ship?

You should be able to solve this.

On average, how many times do you need to roll a die before all six different numbers have turned up?

Apparently pic related is a thing.
Post math terminology that tripped you up at one point.

Mine is [tex: \frac{8}{23}].

The Android calculator app has a cool feature you might not have noticed: swipe left on the result and you get as many decimals as you want.
/watch?start&v=CKPRhaaAAtk

We have n savings boxes, with different keys. A Jew locks the boxes and throws the keys into them at random, putting one key in each box. A Turk breaks open k boxes. What is the probability that the Turk is now able to open all the rest?

Recently I've started reading into λ-calculus mostly with the pdf attached and some smaller reading into the essay of Kurt Godel on recursive functions and Alan Turing on Turing machines. And I have a few questions

1, So as the book mentions and proves in appendix A3, that simply typed lambda calculus will result in SC (strongly computable) and SN (strongly normalizable) terms. And there are clear rules of inductions under [tex: \lambda\beta\rightarrow], we can determine within a finite amount of time whether a given λ term can or cannot be typed. The logical conclusion from this is that there are λ terms that are not typeable under simply typed λ-calculus, but are SC and SN (or WN).
One such example that comes to mind is (λx.xx) (λx.x) that has one path of normalization ending in (λx.x), but I'm not too familiar with the subject to make this call.


2, The Church-Rosser theorem says that
>[tex: \beta]-nf of a given term is unique modulo [tex: \equiv_\alpha]
So Ω being the non-reducing, minimal function (λx.xx) (λx.xx), and L be (λx.xxy)(λx.xxy). How does Church-Rosser theorem apply to:
(λx.y) Ω
or
(λx.y) L
For both reduction takes you straight towards the 'y' atom, but there is an infinite chain of reductions for the other term, at any point of which, it can resolve back into the constant 'y'. What are we supposed to make of this? Or I suppose for any given reduction you can make, there will in fact, always exist another reduction you can make.


3, Turing machines and λ-calculus are supposed to be equivalent, at least to a naive outsider. However, there were valid questions raised when I looked into an algorithm for converting between the two. The gist is that Turing machines cannot take anything else but an input (encoded in) natural numbers, making it unable to accept non-encoded higher order functions, but at the same time being able to run analysis on the inside of encoded functions. Something which λ-calculus is unable to do. Making some set of programs seemingly non-translatable between the two. Which wasn't supposed to happen.
Also on this note, if there aren't clear conversion algorithms between Turing machines and λ-calculus, are there conversion algorithms between actual code and Turing machines. A quick search turns up nothing, but I remember that Brain***** is an extension of P'' which is explicitly is a Turing machine. But other then Brain*****, how would one go about, for example translating a given scope within a Lua program?

4, This is a bit overused and meme-y, but why are there so few uses to λ-calculus? It's used inside the field of mathematics, obscure linguistic research, and programming languages that nobody in the wider population uses. (no offense to anyone, being a Haskell user myself) Or is this just the engineers taking over the mathematicians to do their job and not care about century old weird modes of computation that a few nerds and university professors in the 60s started using.

Six pennies are arranged on a flat surface as shown in the top picture. The problem is to move them into the formation depicted at bottom in the smallest number of moves. Each move consists in sliding a penny, without disturbing any of the other pennies, to a new position in which it touches two others. The coins must remain flat on the surface at all times.

This is a thread to discuss the HoTT book [spoiler: https://hott.github.io/book/hott-online-1353-g16a4bfa.pdf]. It doesn't require any prerequisite materials as it aims to be an informal (yet informative, we just abstract away context dependency and a few details) introduction to homotopy type theory :D I suggest we start with chapter 1 which is about MLTT (martin löf's dependent type theory).

How do you understand mathematics, which is an abstract science? I feel strange. I do not mean the use of mathematics in physics and other fields. I mean mathematics as an abstract science, for example, what will we benefit from the absolute value of x

Does anybody know other science related boards on altchans (ideally maths and physics related)? I feel like i'm missing out, but probably there isn't much demand. I guess reddit has some nice subs for that, but i don't feel like creating an account and grinding for upvotes. So far enjoying /math/.

Have you ever used the interactive theorem prover Lean?
https://leanprover.github.io/
Thinking about giving it a try. I'm already familiar with Coq, but having native support for quotient types sounds nice.

It seems most people interact with Lean through either VS Code or Emacs. Any thoughts on VS Code? (Obviously I'd want to turn the telemetry off if I went with it.)

How to learn precalculus fast?

why are people always saying this word multiplie? what does it mean? what's so special about just getting multiple of something
if i take 3 dollar bills did i multplie the dollar bill?

How many ways are there to write the numbers 0 through 9 in a row, such that each number other than the left-most is within one of some number to the left of it?

There is a simple procedure by which two people can divide a cake so that each is satisfied he has at least half: One cuts and the other chooses. Devise a general procedure so that n persons can cut a cake into n portions in such a way that everyone is satisfied he has at least 1/n of the cake.

Mathematicians have not yet succeeded in finding a formula for the number of different ways a road map can be folded, given n creases in the paper. Some notion of the complexity of this question can be gained from the following puzzle invented by the British puzzle expert Henry Ernest Dudeney.

Divide a rectangular sheet of paper into eight squares and number them on one side only, as shown in the top drawing. There are 40 different ways that this "map" can be folded along the ruled lines to form a square packet which has the" 1" square face-up on top and all other squares beneath. The problem is to fold this sheet so that the squares are in serial order from 1 to 8, with the 1 face-up on top.

If you succeed in doing this, try the much more difficult task of doing the same thing with the sheet numbered in the manner pictured at the bottom of the illustration.

A group of airplanes is based on a small island. The tank of each plane holds just enough fuel to take it halfway around the world. Any desired amount of fuel can be transferred from the tank of one plane to the tank of another while the planes are in flight. The only source of fuel is on the island, and for the purposes of the problem it is assumed that there is no time lost in refueling either in the air or on the ground. What is the smallest number of planes that will ensure the flight of one plane around the world on a great circle, assuming that the planes have the same constant ground speed and rate of fuel consumption and that all planes return safely to their island base?

Can you prove or find a counterexample?

Every odd prime is the sum of two perfect squares if and only if its remainder when divided by 4 is 1.

For example,
[tex:5 = 1^2 + 2^2] [tex:13 = 2^2 + 3^2] [tex:17 = 1^2 + 4^2]
[tex:29 = 2^2 + 5^2] [tex:37 = 1^2 + 6^2] [tex:41 = 4^2 + 5^2]
3, 7, 11, 19, 23, 31, 43, and 47 have remainder 3 when divided by 4 and cannot be written as the sum of two perfect squares.

There is an imageboard that you have recently become aware of (pic unrelated). You don't know its URL, but you do have a small number of screenshots of individual posts. Like on most boards, the posts are numbered sequentially starting from 1. The screenshots show this number, but they don't show the date of the post. How would you estimate the total number of posts made on the board?

In a room stand n armed and angry people. At each chime of a clock, everyone simultaneously spins around and shoots a random other person. The persons shot fall dead and the survivors spin and shoot again at the next chime; eventually, everyone is dead or there is a single survivor.

As n grows, what is the limiting probability that there will be a survivor?

This may be a bit bloggy but I was watching the videos in P1442 and he started talking about ideals so I guess I need to learn a bit about rings to understand it better.

Rings are generalizations of the integers. Rings must satisfy
(a+b)+c=a+(b+c), a+0=a, a+(-a)=0, a+b=b+a,
(a*b)*c=a*(b*c), a*1=a, 1*a=a, a*(b+c)=a*b+a*c, (a+b)*c=a*c+b*c
and in some definitions also
a*b=b*a
but don't need to have a division operation.

One example of a ring besides the integers (and rationals, reals, complexes) themselves are the Gaussian integers, which are the complex numbers
a+bi
where a and b are integers. Another group of examples is polynomials where the coefficients are members of some ring.

In rings you can generalize the idea of multiples and factors. For example, pic related is a plot showing the Gaussian integers, and I've plotted red circles at the numbers that are multiples of 2+i. (And I'm sure you were all forced to factor polynomials in high school.)

I'll post more illustrations I make for myself playing around with rings. If you have favorite textbooks about ring theory, interesting facts about rings, questions or corrections, please join in.

Which version of division with remainder do you think is best and why?
[bold: truncation division] (quotient rounded toward zero)
> 10 ÷ 3 = 3 r 1
>-10 ÷ 3 = -3 r -1
> 10 ÷ -3 = -3 r 1
>-10 ÷ -3 = 3 r -1
[bold: floor division] (quotient rounded down)
> 10 ÷ 3 = 3 r 1
>-10 ÷ 3 = -4 r 2
> 10 ÷ -3 = -4 r -2
>-10 ÷ -3 = 3 r -1
[bold: "Euclidean" division] (0 ≤ remainder < |divisor|, as described in PDF)
> 10 ÷ 3 = 3 r 1
>-10 ÷ 3 = -4 r 2
> 10 ÷ -3 = -3 r 1
>-10 ÷ -3 = 4 r 2
[bold: or something else?]

Veritasium recently released a pop-math video on p-adic numbers (and n-adic numbers, the case where the base is not prime). Technically it would be more correct to say he covered p-adic/n-adic [bold:integers], since he ignored the cases where the numbers have finitely many digits after the radix point.

https://www.veritasium.com/videos/2023/6/6/the-most-useful-numbers-youve-never-heard-of
/watch?v=tRaq4aYPzCc

If you've studied a lot of math you're probably already aware of p-adics. Otherwise it may seem kind of schizo. But in the video he shows how to add, negate, and multiply n-adic integers.

[bold:Challenge]: Can you prove that the algorithms shown in the video obey the familiar rules of working with numbers
>(a+b)+c=a+(b+c), a+0=a, a+(-a)=0, a+b=b+a,
>(a*b)*c=a*(b*c), a*(b+c)=a*b+a*c, (a+b)*c=a*c+b*c
>a*1=a, 1*a=a, a*b=b*a
or in other words, that the n-adic integers are a commutative ring with identity?

Three cards, an Ace, King, and Queen, lie face-up on a desk in some or all of three marked positions ("left," "middle," and "right"). If they are all in the same position, you see only the top card of the stack; if they are in two positions, you see only two cards and do not know which of the two is concealing the third card.

Your objective is to get the cards stacked on the left with Ace on top, then King, then Queen on bottom. You do this by moving one card at a time, always from the top of one stack to the top of another (possibly empty) stack.

The problem is, you have no short-term memory and must, therefore, devise an algorithm in which each move is based entirely on what you see, and not on what you last saw or did, or on how many moves have transpired. An observer will tell you when you've won. Can you devise an algorithm that will succeed in a bounded number of steps, regardless of the initial configuration?

How do I even learn mathematics

What do you think of the axiom of choice? Do you take mathematical constructs that require the axiom of choice to show they exist as seriously as other mathematics? What about nonconstructive mathematics in general?

PDF related, he argues
>Thus the problem with Zermelo’s axiom of choice is not the existence of the choice function but its extensionality, and this is not visible within an extensional framework, like Zermelo-Fraenkel set theory, where all functions are by definition extensional.
(A function is extensional if it maps equivalent inputs to equivalent outputs.)

After the revolution, each of the 66 citizens of a certain country, including the king, has a salary of one dollar. The king can no longer vote, but he does retain the power to suggest changes -- namely, redistribution of salaries. Each person's salary must be a whole number of dollars, and the salaries must sum to 66. Each suggestion is voted on, and carried if there are more votes for than against. Each voter can be counted on to vote "yes" if his salary is to be increased, "no" if decreased, and otherwise not to bother voting.

The king is both selfish and clever. What is the maximum salary he can obtain for himself, and how long does it take him to get it?

1 + 1/2 + 1/4 + 1/8 + ... converges to 2 in the reals.
1 + 2 + 4 + 8 + ... doesn't converge in the reals but it does converge to -1 in the 2-adics.
Is there an extension to the rationals in which 1 + 2 + 3 + 4 + ... converges to -1/12?

A thrackle is a drawing on the plane consisting of vertices (points) and edges (non-self-intersecting curves) such that:

• every edge ends in two different vertices, but hits no other vertex; and
• every edge intersects each other edge exactly once, either at a vertex or by crossing at an interior point.

Is there a thrackle with more edges than vertices?

Discussion and updates and thoughts about this book as people study it.

***** mode: find and prove the exact value of [tex:\alpha], trigonometry allowed
babby mode: find [tex:\alpha] approximately, anything goes except looking up the answer

Mathematical induction is supposedly the basic idea that underlies all of arithmetic. But many people who have no problem with arithmetic struggle with induction. Are mathematicians overcomplicating arithmetic, or are teachers just shit at explaining induction?

If I were proving the a*b = b*a using a machine with formal logic, it would be easiest to write a proof by induction. But if I were explaining why it was true to a human *****, I would draw a picture like the one attached. Are there any formal logic descriptions of arithmetic that are natural in the sense that they make arguments like the one in the picture easy to write? Alternatively, what's the best way to write the picture's argument as a formal proof in existing systems?

tbh this is a stupid idea

1. The Bluebook app is only available on Windows, Mac OS and iPad (no Linux support)
2. Desmos calculator refused to work on tor (I don't want to use clearnet and tor at the same time!!!!)
3. You need a (((Windows / MacOS))) device to test. Luckily you can borrow (((Chromebook))) from your school.


https://blog.arborbridge.com/sat-will-become-fully-digital-and-shorter-by-2024-whats-changing

You should be able to solve this.

These are the best known ways to pack a given number of identically-sized squares into a larger square.
https://erich-friedman.github.io/packing/squinsqu/

Consider two random matrices [tex: C\in\mathbb{R}^{n×n}] and [tex: x\in\mathbb{R}^{n×1}].
Assuming that the first and second moments of C and x (or any additional moment, if required) are known and that C is invertible, how can one find the first moments of [tex: C^{-1} x]?

What is the least number of pieces that a 13x13 square can be cut into which can be reassembled into two squares, one 12x12 and the other 5x5?

thread to discuss the textbook, its exercises, rant about probability theory, and journal as you go through the chapters one by one

Imagine a die with a side for every positive integer, each equally likely to be the outcome if the die is rolled. What is the probability of rolling a 6?

Have you studied any systems of logic other than classical logic? Have you found them useful?

You have an ordinary 8 × 8 chessboard with red and black squares. A genie gives you two “magic frames,” one 2 × 2 and one 3 × 3. When you place one of these frames neatly on the chessboard, the 4 or 9 squares they enclose instantly flip their colors.

Can you reach all [tex: 2^{64}] possible color configurations?

Post interesting math books.

ONAG is about the Field of "surreal numbers" which adds to the real numbers a whole bunch of infinitely large and small numbers. It also talks about the game theory that inspired them (I found it helpful to skip forward and read some of this section first).

Marvin is playing a solitaire game with marbles. There are n bowls (for some positive integer n), and initially each bowl contains one marble. Each turn, Marvin may either

remove a marble from a bowl, or
choose a bowl A with at least one marble and a different bowl B with at least as many marbles as bowl A, and move one marble from bowl A to bowl B.

The game ends when there are no marbles left, but Marvin wants to make it last as long as possible. What is his optimal strategy?

You and a friend are discussing how you choose four-digit PINs. You establish that neither of you would ever use the digit 0.

“I like to choose four different random digits,” you say.

“I like to choose three different random digits,” they reply, “so one of the digits is used twice.”

Which strategy gives the largest pool of possible four-digit PINs?

Here's a nice paper about the disease that has infected academic statistics.

Caitlin wants to draw n straight line segments, without lifting her pencil off the paper and without retracing her path, so that each segment crosses exactly one other segment (not counting intersections at vertices) and she ends up back where she started.

a. Show how she can do this for n = 6. (Draw a picture!)
b. For which n can it be done, and for which n is it impossible? Prove your answer.

What is the value of
[tex: \lfloor (\sqrt{1.000003} - 3\sqrt{1.000002} + 3\sqrt{1.000001} - 1)^{-1/3} \rfloor]?
(Your calculator will probably do the problem, but it may not get it right.)

[tex: \lfloor x \rfloor] means the floor function, the greatest integer not exceeding x.

A farmer sold 108 pounds of produce that consisted of z pounds of zucchini and c pounds of cucumbers. The farmer sold the zucchini for $1.69 per pound and the cucumbers for $0.99 per pound and collected a total of $150.32. Which of the following systems of equations can be used to find the number of pounds of zucchini that were sold?

What do you think about supertasks (tasks with an infinite number of steps, where we're interested in what happens after you finish the infinite steps)? Why do they so often lead to paradoxes? Are mathematicians right to avoid them? Does Zeno's paradox about Achilles and the tortoise mean we have to grapple with supertasks anyway?

Two regular blinkers begin with synchronized blinks at time 0, and afterwards there is an [bold: average] of one blink per minute from the two blinkers together. However, they never blink simultaneously again (equivalently, the ratio of their frequencies is irrational).

Prove that after the first minute (from time 0:00 to time 0:01) there is [bold: exactly one blink] in every interval between time t minutes (t an integer) and time t + 1!

Three sisters combine their money to buy a mystery box full of an unknown amount of comic books. The box arrives at the sisters’ house one night, and they agree to store it in their *****er brother’s room and not open it until the next morning.

But that night, one sister can’t sleep, so she decides to sneak in and open the box. Looking inside, she finds that she can split the comic books into three equal shares, plus one extra comic. She sees her brother awake and spying, so she hands him the extra comic, takes her own share of one-third of the remaining comic books, and goes back to sleep.

An hour later, the second sister does the exact same thing. She opens the box and sees that the comics can be split into three equal shares, plus one extra comic. She bribes the spying little brother with a comic, takes one-third of the remaining books, and goes back to sleep.

Then, an hour after that, the third sister does the same thing yet again, bribing the spying little brother with a comic, taking one-third of the remaining books, and returning to bed.

In the morning, none of the sisters want to reveal that they’ve already taken any comics. So they each pretend to be surprised when they look in the box. They dump out the comics and realize that the number of books can easily be split into three equal shares along with one extra comic, which they gave to their happy little brother.

What’s the smallest possible number of comic books that could have been in that mystery box when it first arrived? And how many comics does each sibling have?

A baker’s dozen (thir*****) bagels have the property that any twelve of them can be split into two piles of six each, which balance perfectly on the scale. Prove that all the bagels have the same weight.

The hour and minute hands of a clock are indistinguishable. How many moments are there in a day when it is not possible to tell from this clock what time it is?

If only two immortal humans distributed randomly on the Earth's landmass survived a collapse of the human population, what would be the most likely time frame for them to find each other?

Let's assume all radio systems were destroyed or they just didn't know how to use them or weren't listening to a long range radio for messages.

How would you estimate or predict the range of time it would likely take for them to find each other?

And what is your prediction?

A colony of chameleons currently contains 20 red, 18 blue, and 16 green individuals. When two chameleons of different colors meet, each of them changes his or her color to the third color. Is it possible that, after a while, all the chameleons have the same color?

You want to connect the points A, B, C and D to each other by drawing the connection with a marker. But you want to use the minimum amount of ink of the marker.

What is the shape that you will draw?

Fif***** people are trapped aboard a ship that's going to sink in exactly 20 minutes. Their only chance for survival is the five-person life raft stowed on their vessel. To make matters worse, the waters around the ship are teeming with man-eating sharks, so swimming to safety is out of the question.

A round-trip to the nearest island and back to the ship takes nine minutes on the raft. How many people will live to see dry land?

Imagine a digital clock like the one shown below. How many times will the clock display three or more of the same number in a row over the course of one day?

In case you were wondering, the clock in this puzzle displays time on a 12-hour scale, not on military time.

From which path should one learn?

Can you solve it?

How would you explain to an elementary school student why n + 0 = n?
How would you explain mathematical induction to a high school student?
And how can formal mathematical proofs be made more similar to the way a non-autistic person would explain things?

Observe that

[tex: (2 − 1)! + 1 = 2^1],
[tex: (3 − 1)! + 1 = 3^1],
[tex: (5 − 1)! + 1 = 5^2].

Are there any other primes p such that (p − 1)! + 1 is a power of p?

Sonny and Cher play the following game. In each round, a fair coin is tossed. Before the coin is tossed, Sonny and Cher [bold: simultaneously] declare their guess for the result of the coin toss. They win the round if both guessed correctly. The goal is to maximize the fraction of rounds won, when the game is played for many rounds.

So far, the answer is obviously 50%: Sonny and Cher agree on a sequence of guesses (for example, they decide to always declare “heads”), and they can’t do any better than that. However, before the game begins, the players are informed that just prior to the first toss, Cher will be given the results of all the coin tosses in advance! She has a chance now to discuss strategy with Sonny, but once she gets the coinflip information, there will be no further opportunity to conspire. Can you help them get their winning percentage up over 70%?

A combination lock with three dials, each numbered 1 through 8, is defective in that you only need to get two of the numbers right to open the lock. What is the minimum number of (three-number) combinations you need to try in order to be sure of opening the lock?

Post mathematical proofs you like.

A large rectangle in the plane is partitioned into smaller rectangles, each of which has either integer width or integer height (or both). Prove that the large rectangle also has this property.

A bus left the Scarlet Devil Mansion. Three people boarded at the start. At Hakugyokurou, one got off, and half a person boarded. At Yakumo's house, two people left. So how many passengers in total?

Ashford, Baxter, and Campbell run for secretary of their union, and finish in a three-way tie. To break it, they solicit the voters' second choices, but again there is a three-way tie. Ashford now steps forward and notes that, since the number of voters is odd, they can make two-way decisions; he proposes the voters choose between Baxter and Campbell, and then the winner could face Ashford in a runoff.

Baxter complains that this is unfair because it gives Ashford a better chance to win than either of the two other candidates. Is Baxter right?

Someone on 4chan once asked why the standard division algorithm starts at the left (most significant digit) of the number whereas the algorithms for addition, subtraction, and multiplication start at the right (least significant digit). Thinking about it, you can apply more or less the standard long division algorithm starting from the right so long as the last digit of the divisor is relatively prime with 10 and there is no remainder:

5634
-----
7 |39438
28
--
41
21
--
92
42
--
35
35
--
0

If there is a remainder, you could continue the division, like so:

...142857143
--------------
...00000000001|7
21
--
...98
28
--
...97
7
--
...99
49
--
...95
35
--
...96
56
--
...94
14
--
...98
28
--
...97
7
--
...99

I don't know if this is how they came up with the idea originally, but this seems to be motivating constructing p-adics.

What's the smallest prime factor of [tex: \frac{2022^{2022} + 6}{6}]?

Does the equation, [tex: x^2 + y^3 = z^4] have solutions in prime numbers? Find at least one if yes, give a nonexistence proof otherwise.

from Vlad Mitlin via https://www.math.utah.edu/~cherk/puzzles.html

Do you have any traumatic /math/ related stories in your life?
I failed math once in my high school.
Me and another friend who sat in front of me during exams copied each other, and we both failed together.
Usually, we could beg a few marks, and shed some tears, to make the teacher pass us, but this asshat teacher was a new guy, and he taught like shit as well which could be a reason why I failed in the first place, and he was quite stubborn and refused to pass us off.
I ended up crying. He later ended up getting admitted after a serious complication and quit.

Cute fact: Suppose we have two regular polygons, both with the same side lengths, with one having twice as many sides as the other. Then we can add together the apothem (distance from the center to the midpoint of a side) and the circumradius (distance from the center to a vertex) of the smaller polygon to get the apothem of the larger polygon. It works because the red triangle in pic related is isosceles. Calculating the circumradius by the Pytha*****an theorem, then repeating the whole process for shapes with more and more sides, we end up with a relatively easy-to-explain (I hope) method for calculating pi.

Without looking up the proof, can you prove it?

During World War II, Russian cities near the front were blacked out. Once when it was time to darken the windows, schoolboy Vasya's parents could not find a shade for a window 120 by 120 units. All that was available was a rectangular sheet of plywood. Its area was correct, but it was 90 by 160.

Vasya picked up a ruler and drew quick lines on the plywood. He cut it into two parts along the lines he had drawn. With these parts he made a square covering for the window. How?

Each morning, your fairy godmother appears and gives you a chance to play a game. In this game, she deals 10 cards face down. Nine of the cards are winners, and one card is a loser. If you pick a winning card, you get a prize. You can then either take your prize and walk away or play again for the chance to win a second prize. But if you lose on that second play, you walk away with nothing and the game is over for the day. Each time you succeed, she invites you to play again under the same conditions (win yet another prize or lose everything).

What strategy maximizes the average number of prizes you win each day? And what is that average?

Extra credit: Suppose your fairy godmother deals N cards each day (instead of 10 cards), with N−1 winning cards and one losing card. Now what is your strategy, and how many prizes do you expect to win each day?

Sir Hugh's ***** kinswoman and ward, Lady Isabel de Fitzarnulph, was known far and wide as "Isabel the Fair." Amongst her treasures was a casket, the top of which was perfectly square in shape. It was inlaid with pieces of wood, and a strip of gold ten inches long by a quarter of an inch wide.

When ***** men sued for the hand of Lady Isabel, Sir Hugh promised his consent to the one who would tell him the dimensions of the top of the box from these facts alone: that there was a rectangular strip of gold, ten inches by [tex: \frac{1}{4}]-inch; and the rest of the surface was exactly inlaid with pieces of wood, each piece being a perfect square, and no two pieces of the same size. Many ***** men failed, but one at length succeeded. The puzzle is not an easy one, but the dimensions of that strip of gold, combined with those other conditions, absolutely determine the size of the top of the casket.

00 or 0?

The distance between the towns A and B is 1000 miles. There is 3000 apples in A, and the apples have to be delivered to B. The available car can take 1000 apples at most. The car driver (or perhaps someone accompanying him) has developed an addiction to apples: when he has apples aboard he (or she) eats 1 apple with each mile made. Figure out the strategy that yields the largest amount of apples to be delivered to B.

Generalize the strategy for an arbitrary amount of apples.

from: https://www.math.utah.edu/~cherk/puzzles.html

Just noticed something interesting. Can you prove it or find a counterexample? Any interesting generalizations?

The Fibonacci numbers are given by
[tex: F_0 = 0]
[tex: F_1 = 1]
[tex: F_{n+2} = F_n + F_{n+1}].

For even [tex: n \ge 2],
[tex: \arctan(1/F_n) = \arctan(1/F_{n+1}) + \arctan(1/F_{n+2})].

Sometimes I tutor ***** in math, and I see that a lot of their homework problems are extremely straightforward problems intended to exercise a particular technique they just learned. This is very different from problems people encounter in real life, where the first step is to figure out how to approach the problem, and where problems may involve multiple steps combining knowledge from different areas of math. So I thought it would be useful to collect more realistic problems that ***** could practice. Any level of difficulty and prerequisite knowledge is welcome. If you have any good books full of problems to recommend, that would be helpful too.

Pic related, a problem someone posted on 4chan /sci/ a while back as an example of math they use in their job.

Where did the extra area come from?

Find a 10-digit number where the first digit is how many zeros in the number, the second digit is how many 1s in the number etc. until the tenth digit which is how many 9s in the number. Is the solution unique? What happens for an n-digit number in base n?

Can you find the minimum number of quarter face turns needed to
>swap two pairs of corner pieces
>swap three corner pieces
>swap two pairs of edge pieces
>swap three edge pieces
>swap a pair of corner pieces and a pair of edge pieces
>change the orientation of two corner pieces
>change the orientation of two edge pieces
?

Anna loves multiples of 11, but her friend Jane is not quite so keen. One day, Anna is flipping idly through the yellow pages (remember those?), which is full of 10-digit numbers. She notices that every 10-digit number seems to have an interesting property: It is either a multiple of 11, or it can be made a multiple of 11 by changing a single digit. For example, there are several ways to make the 10-digit number 5551234567 into a multiple of 11, such as changing the first digit to 4.

This gets the two friends wondering: Does every counting number have this property? Either prove it’s true for every number, or find the smallest counting number that is not a multiple of 11 and cannot be made a multiple of 11 by changing one digit.

Problem, Archimedes?

Everybody should be able to solve this.

Find three square numbers in arithmetic progression. Can there be four such?

How many squares are there? If you guess incorrectly you're a low IQ *****. Based on statistics I've witnessed, ~98% of all people are low IQ *****s.

MAGIC SQUARES have intrigued mathematicians for more than 2,000 years. In the traditional form the square is constructed so that the numbers in each row, each column and each diagonal add up to the same total. However a magic square of an entirely different type is pictured in Figure 8. This square seems to have no system: the numbers appear to be distributed in the matrix at random. Nevertheless the square possesses a magical property as astonishing to most mathematicians as it is to laymen.

A convenient way to demonstrate this property is to equip yourself with five pennies and 20 little paper markers (say pieces of paper matches). Now ask someone to pick any number in the square. Lay a penny on this number and eliminate all the other numbers in the same row and in the same column by covering them with markers.

Ask your spectator to pick a second number by pointing to any uncovered cell. As before, put a penny on this number and cover all the others in the same row and column. Repeat this procedure twice more. One uncovered cell will remain. Cover it with the fifth penny.

When you add the five numbers beneath the pennies — numbers chosen seemingly at random — the total is certain to be 57. This is no accident. The total will be the same with every repetition of the experiment.

If you enjoy solving mathematical puzzles, you may wish to pause at this point to analyze the square and see if you can discover its secret yourself.

>from Martin Gardner's _Hexaflexagons and other mathematical diversions_

Consider any triangle ABC such that the midpoint P of side BC is joined to the midpoint Q of side AC by the line segment PQ. Suppose R and S are the projections of Q and P respectively on AB, extended if necessary. What relationship must hold between the sides of the triangle if the figure PQRS is a square?

problem found in https://www.mathematrucker.com/problems_1975-1979.pdf
originally published in the Pi Mu Epsilon Journal
diagram is my own, any mistakes in it are my fault

One side of a right-angled triangle is 48. Find ten pairs of whole numbers which may represent the other two sides.

While walking down the street one day, you notice people gathering around a woman playing the shell game. With each game, she places a ball under one of three cups, and then swaps the positions of pairs of cups several times before asking passersby which cup they think the ball is now under.

You have it on good authority that she is playing fairly, performing all the moves in plain sight, albeit too fast for you to track precisely which cups she’s moving. However, you do have one additional key piece of information — every time she swaps cups, one of them has the ball. In other words, she never swaps the two empty cups.

When it’s your turn to guess, you note which cup she initially places the ball under. Then, as she begins to swap cups, you close your eyes and count the number of swaps. Once she is done, you open your eyes again. What is your best strategy for guessing which cup has the ball?

bonus points for elegance

another puzzle

Can you solve it?

P2585
I wonder if there's a way to automatically convert these nice simple function definitions from non-standard analysis into constructive analysis functions that you can actually do computations with.

P4379
I guess so. I don't know how to come up with a fun and appropriate problem. I want to do something with differential equations, and for it to involve more than one kind of problem solving.
I left one graph off the attached plot image. The same gnuplot but with the left-off plot on it is the goal. The hint as to what the system of equations is, is -of-Arabia.

Here's my gnuplot:
set term png transparent truecolor size 2048,2048
set output "mystery.png"
set nokey;
plot "a1.out" using 1:3 with lines,\
"a2.out" using 1:2 with lines,\
"b1.out" using 1:3 with lines,\
"b2.out" using 1:2 with lines,\
"c1.out" using 1:3 with lines,\
# "c2.out" using 1:2 with lines,\

Easy? Stupid? Abstruse? Too much work? Not technical enough?

i just dont get this, say i have a 100 euros in a stock and that stock goes up by 10% one day
i now have 110 euros in that stock
now the next day the stock goes up another 10%
10% of 110 is 11 so now i have 121 euros in that stock
so 2 times 10% is 121
but if that stock had gone up 20% on the first day and zero on the second i would only have 120
where did the extra euro come from

Are these statements consistent with each other in intuitionistic logic?
Is there a way to prove if they are or aren't?

It's popular for mathematicians to roleplay that they are working in ZFC, a theory of sets whose members are sets. What would modeling mathematics in list theory look like? I'm imagining a theory where the objects are lists whose members are lists. Lists would be like sets, but would allow repeated elements, and the elements would have an order. Infinite lists would of course be allowed. What axioms would be appropriate for list theory? How would doing mathematics in list theory compare to working in set theory?

Mr. Smith and his wife invited four other couples for a party. When everyone arrived, some of the people in the room shook hands with some of the others. Of course, nobody shook hands with their spouse and nobody shook hands with the same person twice. After that, Mr. Smith asked everyone how many times they shook someone's hand. He received different answers from everybody. How many times did Mrs. Smith shake someone's hand?

Elchanan first presented a certain puzzle to his probability class a couple of years ago, when he was giving examples of conditional expectation computation. On the spot, he came up with a problem and derived the solution. “Then I looked at it and asked if it made sense. It did not, as my intuition very strongly warned me that the answer I derived was off. It took me some time to recover and figure out why my intuition was wrong.” And the answer has stumped many since it was presented on Gil Kalai’s blog “Combinatorics and More.” “I am somewhat surprised that it got so popular,” Elchanan added.

Test your intuition:

You roll a die until you get 6. What is the expected number of rolls (including the roll giving 6) conditioned on the event that all rolls gave even numbers?

The expected number of throws:

a. 1 b. 3/2 c. 2 d. 5/2 e. 3 f. 7/2 g. 4

Can you show the segments connecting the centers of the squares have equal length?

1. Start by posing the problem of finding the hypotenuse of a right triangle given the lengths of its legs (use convenient concrete numbers, such as 3 cm and 4 cm). Are they stuck? Draw a square on the hypotenuse and ask how to find the area of the square. Point out that if we knew the area of the square by some other means, we can work backwards to figure out the length of the hypotenuse. Now draw the usual square around the square. Have the ***** compute the area of the inner square by subtracting the area of the triangles from the area of the outer square.

2. Explain that there's an easier way to get the same result. Draw the rearranged diagram featuring the squares of the two legs. Have them work out the sum of the areas of the two squares. See if they can explain why the results are the same. If not, point out that it's the same outer square minus the same triangles.

3. Now it's time to write down the formula a²+b²=c² and explain it in terms of the areas of the squares.

Started reading this short book on model theory I came across. It's a field I keep hearing about bizarre meta-mathematical results from, which I'm hoping to understand a bit better. Also apparently model theory is the origin of non-standard analysis.

Six circles of radius r are inscribed in a rectangle with dimensions L by l as shown. What are L/r and l/r?

As everyone knows, there is a one-to-one correspondence between the rational numbers plus ∞ and the rational points on the circle (if you do not know this, see the first video in P1442). Furthermore, all non-negative rational numbers can be enumerated in simplest form by starting with 0/1 and 1/0 and repeatedly performing idiot addition aka the mediant M(a/b,c/d) = (a+c)/(b+d), like so:

0/1 1/0
0/1 1/1 1/0
0/1 1/2 1/1 2/1 1/0
0/1 1/3 1/2 2/3 1/1 3/2 2/1 3/1 1/0

Simply flip the signs to get the negative rational numbers also. By connecting the points these rational numbers correspond to, we can generate a series of inscribed 2ⁿ-gons with rational area which converge to the circle. We can also generate circumscribed 2ⁿ-gons that converge to the circle using the tangent lines at the points. It's probably a terribly inefficient way to compute π, but I'm curious how it will work out.

Do you use SageMath?

P3661 had some notion of [1] f(z; t) = f(z-1; t-1) + f(z-j; t-1)
(8 8 0)=1, t=5:
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 5 1 0
0 0 0 0 0 0 10 4 1 0
0 0 0 0 0 10 6 3 1 0
0 0 0 0 5 4 3 2 1 0
0 0 0 1 1 1 1 1 1 0
0 0 0 0 0 0 0 0 0 0
I wanted to connect this a bit differently as well. Like what if t=0, (5 5 0)=1, t=4 below.
[2] f(z; t) = f(z-1; t-1) + f(z-j; t-1) + f(z+1; t-1) + f(z+j; t-1)
0 0 0 0 0 0 0 0 0 0
0 1 4 10 16 19 16 10 4 1
0 4 13 31 46 55 46 31 13 4
0 10 31 77 114 139 114 77 31 10
0 16 46 114 162 200 162 114 46 16
0 19 55 139 200 249 200 139 55 19
0 16 46 114 162 200 162 114 46 16
0 10 31 77 114 139 114 77 31 10
0 4 13 31 46 55 46 31 13 4
0 1 4 10 16 19 16 10 4 1
I guess it's completely described by [2] and is just
[3]
0 1 0
A *(4-1) 1 0 1
0 1 0
With *(4) being repeated convolutions. (How would you type that normally?). A is like zeros(10 10), A[5 5]=1. Which is not that exciting, though linear filtering is cool I guess.
Instead I had the idea of doing game of life, then slowly adding specific next-nearest-neighbor exceptions to the rules to try to control the world a little bit.

Complex number are undeniable proof that the mathematically ideal world is 2D. Prove me wrong.
[spoiler: Protip: you can't.]

To what extent are the Internet arguments about 0.999... legitimate, and to what extent is it just people being contrarian?
Are you convinced that 0.999... = 1?

Have you watched any interesting math videos lately? I've been slowly going through a series of lectures on algebraic geometry. Slowly mostly because I find myself pausing a lot to work on the problems he's talking about. I haven't gotten very far yet, but so far it's pretty interesting.

https://www.youtube.com/playlist?list=PL8yHsr3EFj53j51FG6wCbQKjBgpjKa5PX

Draw two lines g and h. Then draw three points A, B, C on line g and 3 points a, b, c on line h. Let X be the intersection of lines Ab and Ba, Y of Ac and Ca, and Z of Bc and Cb. Pappus's theorem says points X, Y and Z are collinear. Can you see why?

Can you cover all the squares with dominos? Each domino covers two adjacent squares, and no part of the domino is allowed to go outside of the squares.

This is a doodle I made of the Gaussian primes.
Have you made any math-related art you'd like to share?

Can you color this map with the least number of colors possible? Each pair of regions that share a boundary must be a different color. Regions that only touch at a corner don't have to have different colors.

For 2×2 * 2×2 matrix multiplication over non commutative rings it is a well know results by Strassen that this can be performed with 7 multiplications. Consider:

p₁ = (x₁₁ + x₂₂)×(y₁₁ + y₂₂)
p₂ = (x₁₁ + x₂₂)×y₁₁
p₃ = x₁₁×(y₁₂ - y₂₂)
p₄ = x₂₂×(-y₁₁ + y₁₂)
p₅ = (x₁₁ + x₁₂)×y₂₂
p₆ = (-x₁₁ + x₂₁)×(y₁₁ + y₁₂)
p₇ = (x₁₂ - x₂₂)×(y₂₁ + y₂₂)

( z₁₁ z₁₂ ) = ( p₁ + p₄ - p₅ + p₇ p₃ + p₅ )
( z₂₁ z₂₂ ) = ( p₂ + p₄ p₁ - p₂ + p₃ + p₆ )

However the minimum number of multiplication for larger matrices is still unknown. This fact about 2×2 matrices is useful as it leads to an algorithm by recursion which allows O(nˡᵒᵍ²⁽⁷⁾) multiplication of n×n matrices. New results for other prime sized square matrices or for powers of two would be extremely useful. New results for non square matrices would also be useful.
(Sorry for the repost)