# 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 ...
3k views

### 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 ...
558 views

### 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 ...
323 views

### 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 ...
74 views

During my regular recreational late night Desmos foolery, I came across this fractal parametric equation: $$x(t)=\sum_{n=0}^\infty \frac{\cos(2^nt+cn)}{2^n}$$ $$y(t)=\sum_{n=0}^\infty \frac{\sin(2^nt+... 0answers 409 views ### Largest consecutive integer using basic operations and optimal digits? If you are first time reading this, you may want to read the summary section last. Solution summary and questions Sequence values If the allowed operations are (+,-,\times,\div) and parentheses (... 0answers 447 views ### What turmite runs the longest before becoming predictable? When looking at 2D Turing machines, many of them eventually become predictable. For example, Langton's Ant, the champion 2-color 1-state turmite, develops a highway after 10,000 steps. Predictable ... 0answers 316 views ### Number of circles in configuration Consider the n^2 lattice points (i, j), where 1 \leq i, j \leq n. Let the number of circles that pass through at least 3 points of this set be C(n). What is a good way to count this? Is there ... 1answer 443 views ### Strategy for board game 2 In this question the following was asked: Alice and Bob are playing the following game: They have a 4 \times 4 empty grid and take turns coloring one square each, starting with Alice, both using ... 0answers 215 views ### Generalization of Gray codes A friend of mine asked me if it was possible - physical difficulties aside - to generate all 32 combinations of raised/lowered fingers by changing status of a fixed number of fingers at every step. ... 0answers 221 views ### Take an m x n grid, and in each box pick two opposite corners at random to connect. What can be said about the resulting pattern? Inspired by the upcoming book 10 PRINT CHR(205.5+RND(1)); : GOTO 10 by Nick Montfort et al., whose title derives from this particular example of emergent behavior. Here's an example: (Note that I'... 0answers 674 views ### Progressive Dice Game (2019.) Edit: Rewriting the question to make it clear. The progressive dice game At the start, you have a fair, regular six sided dice D=(1,2,3,4,5,6). The game is played for n turns. ... 1answer 348 views ### Mathematics of the Ice Bucket Challenge I've been considering the mathematics of the now global ice bucket challenge. Simple model In the simplest incarnation, there is one original seed, who then nominates 3 others, each of which take ... 0answers 197 views ### Rolling Dodecahedron Path I've recently studied rolling polyhedra graphs. A polyhedron is rolled over its edges until it has reached all possible orientations. One more roll puts it back where it started. Here's the 120 ... 2answers 247 views ### Determining finitude or infinitude from a simple geometric construction Playing with a pencil on a checkered sheet I encountered this construction: 1) take a point A on the grid and a point B that is distant from A n=2,3,4... horizontal steps and 1 vertical ... 1answer 493 views ### A non-composite sequences Can you provide a counterexample for a claim given below? Inspired by Puzzle 937 I have formulated the following claim: For any n > 0 let B = p_1 \cdot p_2 \cdot .... \cdot p_n be the ... 0answers 305 views ### Frogs jumping on trees The frog game on a Tree: Default start of the game is placing one frog on each node (vertex). The goal is to move all the frogs to one single node. A single move consists of moving all n frogs from ... 1answer 269 views ### What would you see inside a spherical mirror? Image to build a huge spherical shell made of semitransparent glass, and to cover the internal part with a reflecting material. In such structure some light can enter, and an observer inside it (e.... 0answers 331 views ### Rep-tiles of order 4 and order 9 Rep-tiles are figures which are dissected by the same figures as itself. As you can see, the rep-tile of order 4 is also a rep-tile of order 9 in the above figures: Compare the L-shaped figure at ... 0answers 323 views ### Approximating \pi by an expression of the form \sqrt{\sqrt{ \cdots \sqrt{ n!! \cdots !}}} Here is a problem that appeared as a prize challenge in a periodical for science students, back when I was a student: Find an approximation of \pi formed of the digits 0 through 9, each used at ... 0answers 422 views ### 5 \times 5\; “square additive set” Problem: IBM Research - Ponder This - January 2019 monthly contest (which was closed few days ago) leads to the problem: Find sets A = \{a_1,a_2,\ldots,a_n\}, B = \{b_1,b_2,\ldots, b_m\} such ... 1answer 482 views ### leaf-labelled unordered, rooted binary trees and perfect matchings While playing with findstat.org, I noticed the following coincidence: The number of leaf labelled unordered rooted binary trees with n+1 leaves \{1,\dots,n+1\}, with the leaf labelled 1 at ... 1answer 214 views ### On prime numbers of the form 7\times10^n+69 and the lights out puzzle Consider those natural numbers n such that 7\times10^n+69 is a prime number. The first 15 such numbers are 1, 2, 3, 6, 7, 8, 10, 12, 13, 21, 46, 68, 91, 153, and 366... 0answers 2k views ### Solution manual to Larson's “Problem Solving through Problems” I am working through Larson's "Problem Solving through Problems" (http://math.la.asu.edu/~ifulman/mat194/problem-solving.pdf) but many of the problems have neither solutions nor sources included. Does ... 0answers 241 views ### Olympic number theory problem: is this solution fine and sufficiently well written? Determine all positive integers m such that the ratios$$ \frac{2(5^m+5)}{3^m+1}\quad\text{and}\quad \frac{9^m+1}{5^m+5}$$are both integers. Attempt at a solution: If the ratios are both ... 0answers 256 views ### Are there known pairs of simple numbers equal to huge precision, but not equal strictly? Are there known pairs of numbers a and b, which at first look at them seemed likely to be equal, and after checking up to 10^n decimal places appeared to agree, but suddenly for some n they ... 0answers 207 views ### Heesch numbers in 3D At the Tiling Database there are 3, 20, 198, 1390 (A054361) non-tiling polyominoes of order 7 to 10. In 3D, solids with a particular Heesch number don't seem to be well known. Glenn Rhoads found ... 0answers 197 views ### How can we formalize Jorge Luis Borges' Aleph? Background. Jorge Luis Borges was a post-modern short-story writer of the 20th century, whose stories often invoke a healthy dose of surrealism. One of his works is called The Aleph. In this book, ... 1answer 122 views ### Reversing the digits of 2^n to yield a prime power This question asks about the existence of an n\in\Bbb N such the number obtained reversing de digits of 2^n is a power of 7. The same question with 5 instead of 7 was asked by Freeman Dyson, ... 0answers 293 views ### The Heegner Polynomials What is special about x^3- 6 x^2 + 4 x -2? The 24th power of the real root - 24 is curiously close to two other numbers, one being the Ramanujan constant. There are more of these polynomials ... 0answers 224 views ### How are Trott constants found; are there mathematical results? While reading Steven Finch's wonderful book Mathematical Constants, I encountered the Trott constant which was presented as the real number such that the digits of its decimal expansion are the digits ... 0answers 366 views ### Separating Heavier from the Lighter Balls This was posted Here and received a good answer, solving the general questions in either n or n+1 moves, which is by just half a move on average "less good" than my manual solutions here. Classic ... 0answers 269 views ### Limit approximation for \pi in the four fours puzzle? The four fours puzzle is a recreational math puzzle whose aim is to express whole numbers using four occurrences of the digit 4 and a specified set of operators. A common variety permits the following:... 0answers 327 views ### Name of a certain set I want to know if there is any already-standard way to refer to the sets described as follows. For a set X, let -X = \{-x: x \in X \}; call it the negative of X. Take the set of all primes in \... 0answers 86 views ### Is every permutation of the first k prime bases in p_1^{q_1}+p_2^{q_2}+\dots+p_k^{q_k} a unique sum? Or, is it possible to have two distinct permutations of the first k primes for some k such that using each prime once as a base and once as an exponent to form k terms will yield identical sums? ... 0answers 78 views ### Borromean Weaving Here's a picture by Rashmi Sunder-Raj. Is this topologically equivalent to Borromean Rings? 0answers 75 views ### Is this kind of “Gerrymandering” NP-complete? Consider the following simplified form of "Gerrymandering": You have n^2 squares arranged as an n\times n matrix. Each square is marked with either 0 or 1 which means a "voter preference" ... 1answer 281 views ### Packing problem - how to fit things nicely with just the aspect ratio of the objects Suppose you have a rectangle that is w units wide and h units tall (bounding rectangle). You also have an even ... 0answers 149 views ### Does inscribing a circle, then a triangle, then …, inside of an initial triangle telescope to some “center” of that triangle? Start with a triangle, and construct its inscribed circle. Take the three points where the inscribed circle is tangent to the triangle, and construct a new triangle with those points as the vertices. ... 0answers 174 views ### Magic Cubic Curve Permutations The permutation (-2,9,-4,7,-6,5,-8,3,1) can be considered magical. With their negative values diametrical to 0 at (0,0), a placement of integers begins so that all zero-sum triples form straight ... 0answers 155 views ### Sequential square packings There are various studies for packing sequential squares of size 1 to n. We can try to find the smallest square they will pack into, as in tightly packed squares. We can find the smallest square ... 0answers 480 views ### Oblongs into minimal squares Consider a(n), the minimal number of squares into which the oblong of size (n+1)\times(n) can be divided. What is the behavior of a(n)? The first 379 terms of the oblong square packing sequence ... 1answer 127 views ### For what numbers is a_{b}= b_{a}? (Reference?) A student recently asked me about solutions to the equation$$a_{b} = b_{a}, where the subscript notation $a_{b}$ denotes interpreting the digits of $a$ in base $b$. It turns out there are tons of ...
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 ...