# Questions tagged [recreational-mathematics]

Mathematics done just for fun, often disjoint from typical school mathematics curriculum. Also see the [puzzle] and [contest-math] tags.

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### Can we remove any prime number with this strange process?

This is a little algorithm I made today, which may appear to be quite complex, so I will start with an example. Questions are at the end of the post. The process goes as follows: Start with the first ...
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### Mondrian Art Problem Upper Bound for defect

Divide a square of side $n$ into any number of non-congruent rectangles. If all the sides are integers, what is the smallest possible difference in area between the largest and smallest rectangles? ...
1k views

### Pattern in Pascal's triangle

Updated question This "reverse" pattern can be plotted as a function of a triangle, read by rows: $$T(n,k) = (\delta)^k F\binom{n}{k} \left\lfloor f(t(k)) \right\rfloor ,\delta\in\{1,-1\}.$$...
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### This should be a piece of cake… right?

You probably know the following problem: We have two circular cakes of the same height but unknown and potentially different radii, and we want to cut them into two equal shares. Each cut can only cut ...
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### The Complexity of “The Baby Shark Song”.

This question is just for fun. I hope it's received in the same goofy spirit in which I wrote it. I just had the pleasure of reading Knuth's "The Complexity of Songs" and I thought it'd be hilarious ...
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### Grasshopper jumping on circles

Can we characterize the grasshopper sequence? Let $n\in\mathbb N$ be the number of stones $s\in\{0,1,2\dots,n-1\}=S$ on a circle that the grasshopper can jump on. Let $v(s)$ be the number of times ...
221 views

### Proof request: a collection of sliced squares of size 1 to n can always form a nontrivial rectangle

I'm an active member and challenge writer on Code Golf SE. Here is a challenge of mine, titled Make a rectangle from a collection of (sliced) squares: Task There is a famous formula on the ...
192 views

### Numbers whose decimal digits are the coefficients of its continued fraction form

A curious question recently crossed my mind. And that is can we construct decimal numbers of the form $$\text{"a.bcdefghij..."}$$ where each letter represents a digit $0-9$ (the number may or may ...
749 views

### Prime factor of $2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$

I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below $8*10^9$ Nevertheless, the given number has still ...
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### Determining the strength of an abstract army

Lets imagine we have two armies, represented by lists of pairs of positive numbers, like this: [($attack1$,$defence1$),($a2$,$d2$)...($an$,$dn$)] face each other in combat. The rules of combat are ...
I took prime $131$, squared digits of it and wrote them in natural order as they appear, from left to right, and obtained $191$, then I obtained $1811$ by the same procedure, and then $16411$ and then ...