Puzzles, curiosities, brain teasers and other mathematics done "just for fun".

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1answer
107 views

Discrete Maths question

Show that $S=\{1,3,4,5,9\}$ is a difference set for $\Bbb Z_{11}$. Identify the design produced from $S$ by the sets of the form $S+i$, $i \in\Bbb Z_{11}$.
12
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1answer
198 views

Request for a proof of the following continued-fraction identity

I have been poring over many texts about continued fractions, but none of them seem to be helping me to prove the following beautiful continued-fraction identity (I am nowhere close): $$ ...
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2answers
2k views

How can you construct as many intersections as possible with n lines?

If you have $n$ lines, it seems to be obvious that you can have at most $\frac{n^2-n}{2}$ intersections: $n = 1$: Obviously you need two lines to intersect, so the maximum number of intersections is ...
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1answer
469 views

Puzzle of $N$ men around a table

This was asked to me by a friend. $N$ men sit around a circular table. Man 1 has a sword with him and he kills the Man 2, Man 3 picks up this sword and kills the next person i.e. Man 4. Thus the man ...
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1answer
212 views

How to solve this algorithmic puzzle?

For fixed integers $T\geq G>1$, we say a list $[a_1, a_2,\cdots, a_n]$ is normal if every consecutive sublist $[a_i, a_{i+1}, \cdots a_{i+T-1}]$ of length $T$ has less than $G$ maximal elements. ...
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1answer
148 views

Prove that all combinators must fulfill A x = x for some x, given that M x = x x and composability of any two combinators

I'm working through Raymond Smullyan's "To Mock a Mockingbird" and I'm stuck on the first problem in the combinatory logic section. I'd appreciate hints, but no spoilers please. The problem is ...
12
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1answer
266 views

The number of prime years in a lifetime

$2013$ is not a prime: $3 \times 11 \times 61$. I was born in a prime year, and if I live as expected according to the statistics for U.S. males, I will just reach another prime year, $2027$. That ...
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3answers
157 views

Trouble simplifying a tough equation

I'm having trouble simplifying the following equation. I've tried grouping terms in different ways, but it's not looking any more joyful. Can someone please help with its resolution? Hopefully by ...
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1answer
303 views

How is tessellation defined in Mathematics?

Hi. I am a GCSE student and I am interested in Maths. I read few books on maths and learned some mathematical analysis. I know of convergent series but I would like to know how identical sets(not ...
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3answers
130 views

Pat the Mathemamagician Part 2

Sal the Magician asks you to pick any five cards from a standard deck. You do so, and then hand them to Sal’s assistant Pat. Then you pick one of the five cards, and Pat puts it back into the deck, ...
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0answers
126 views

Is there a way to determine how many solution does “ The hundred Fowls problems” have looking at the coefficients?

I was looking at the different versions of the hundred fowls problems. I came across the problem posed by Chinese mathematicians posed in 5th century,and then Indians in 9th, 10th centuries. Alcuin of ...
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3answers
263 views

Palindrome of numbers from 1-n

A palindrome is a number or word that is the same when read forward and backward, for example, “176671” and “civic.” Can the number obtained by writing the numbers from 1 to n in order (n > 1) be a ...
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1answer
48 views

How to get range of 1 to 10 by index?

I have a collection of lessons divided by levels Each level has 10 chapters. ...
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3answers
8k views

Topology: The Board Game

Edit: I've drawn up some different rules, a map and some cards for playing an actual version of the game. They're available at my personal website with a Creative Commons Attribution 4.0 license. ...
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3answers
534 views

$5n+1$, $3n-1$ problem, smallest repeating cycle and Collatz conjecture

Among the Collatz conjecture we have other "similar" problems that are solved and have repeating cycles. $5n+1$ has the repeating cycle $13, 66, 33, 166, 83, 416, 208, 104, 52, 26$, with a length of ...
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3answers
209 views

What is the value of D here?

Number $S$ is obtained by squaring the sum of digits of a two digit number $D$. If the difference between $S$ and $D$ is $27$, then the two digit number $D$ is? My thoughts: Let the two digit number ...
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2answers
243 views

$N \equiv 3 (\textrm{mod } 4)$ and Collatz conjecture

Can the Collatz conjecture also be interpreted as behaviour, transformation of number of form $N\equiv 3(\textrm{mod }4)$ to the form of $N\equiv 1(\textrm{mod }4)$ Because integers of the form ...
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1answer
235 views

Dissection of a chess board into 4 congruent pieces

Consider a standard $8\times 8$ chessboard where a pawn is placed on each of the squares $d1,d2,d3,d4$ . Dissect the board into $4$ congruent pieces (reflections are allowed) such that each piece ...
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2answers
60 views

Calculating the most efficient number of group payment transactions required

If you have a group of people who purchase items together and split the costs (not always evenly). How can you calculate the most efficient number of transactions required to settle outstanding debt. ...
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4answers
327 views

A logic puzzle from TES: Arena

Its nice when games have riddles hidden in them. While playing TES:Arena, I came across an unusual logical puzzle: There are 3 cells. If Cell 3 holds worthless brass, Cell 2 holds the gold key. If ...
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1answer
377 views

Mathematical Quine

I have recently discovered that I can create letters and any shape I want by hiding parts of curves by making them complex. To generalise if I want $x>a$ then I multiply my function by ...
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2answers
909 views

Useless math that become useful

I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless. My idea is to amend my article with some theories that seemed useless when they are created but ...
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1answer
1k views

In how many ways ( using only whole numbers ) can we divide 49 into 6 parts so that we can obtain any number between 1 to 49?

The series which forms the basis of all the other series is:- 1,2,4,8,16,18. Some other combinations are:- 1,2,3,7,14,22 ; 1,2,4,7,15,20 ; 1,2,4,8,13,21. However, I obtained the basic combination by ...
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2answers
435 views

What is the most unfair set of three nontransitive dice?

In a set nontransitive dice, each die is superior to another die, but is inferior to a third. It is similar to the game of rock-paper-scissors. Here is one example: ...
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3answers
279 views

How to formally model the “hesitation” in the hat-guessing puzzle?

Hua Luogeng (in Chinese, 华罗庚) took a hat-guessing puzzle as an illustration in a booklet focusing on mathematical induction. The following description is a literal translation from Chinese. ...
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1answer
1k views

The $n$ Immortals problem.

I saw this riddle posted on reddit a long time ago, called the "Seven Immortals." In the beginning, the world is inhabited by seven immortals, ageless and sexless, who begin to multiply and ...
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1answer
1k views

If 2 x 6 = 4, 4 x 1 = 7 and 8 x 3 = 1 find the value of 5 x 5. [closed]

Please help me find some logical pattern for If 2 x 6 = 4, 4 x 1 = 7 and 8 x 3 = 1 find the value of 5 x 5.
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1answer
733 views

“$n$-Yahtzee” question

Suppose that you have $n \geq 1$ standard 6-sided dice. If all the dice display the same number, we call this an $n$-Yahtzee. Follow this algorithm: Roll all the dice. Set aside those dice with ...
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1answer
132 views

Monty Hall Application

Driver A comes to a 3 way path junction but is not sure which one to take. Just as he decides to take path 1, a cyclist came by and told driver A all he knows is that he is going on path 3 which would ...
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2answers
316 views

What are good methods for solving Conway's card-stacking puzzle?

Suppose there is a table with three marked spots, $A, B, $ and $C$, on which playing cards can be put, face up. Initially, an ace (1), a deuce (2), and a trey (3) are placed on one or more of these ...
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1answer
227 views

asymptotics of the square indicator sum for subsequences of digits

Introduce the function $g(n)$ defined for positive integer arguments $n$ as the count of squares of positive integers among the numbers that can be formed by taking some subsequence of the digits of ...
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1answer
92 views

finding average price of two lots of shares

Sorry, not a math guy, this might be an overly simple question for the lot of you, but it's a calculation that I'll have to do a lot possibly, so please show me the best way to do it. If I have $1265$ ...
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4answers
131 views

Is there a theorem that disproves this or is this just some made up meaningless thing?

I find this slightly funny. I saw this on a meme:$$\begin{align}a=x\\ a+a=a+x\\ 2a=a+x\\ 2a-2x=a+x-2x\\ 2(a-x)=a+x-2x\\ 2(a-x)=a-x\\ 2=1\end{align}$$ How can these strange algebraic manipulations not ...
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1answer
397 views

finding nearest perfect square for a fractional number

Is it possible to have a clear definition for the nearest perfect square number for a fractional number? For example, let us consider a number 0.004. What is another decimal number closest to it, that ...
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3answers
102 views

What is the smallest number which begins with 7 and if you bring the 7 to the least significant position it becomes a third of the original number?

First I wrote the equation: $7\times 10^2+c_1\times10^1+c_0\times10^0 = 3(100c_1+10c_0+7)$ which becomes $679=290c_1+29c_0$ Then I try fix as many variables as possible. In this first iteration, ...
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2answers
249 views

Is my proof that the medians of a triangle are concurrent valid?

Consider any triangle ABC. Connect the midpoints of each of the three sides. The inscribed triangle is equal to the other three triangles and they are all congruent. It turns out that the medians of ...
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1answer
308 views

How did Euler solve the 4-whole-numbers-adding-up-to-a-perfect-square problem?

So I was watching a video on Leonhard Euler about how he amazingly solved so many difficult problems and one of the many problems that he solved was this: ...
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0answers
123 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 ...
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1answer
141 views

Linear Algebra Recreational Problem

For each positive integer $k$, find the smallest number $n_k$ for which there exist real $n_k$ by $ n_k$ matrices $A_1; A_2; ....; A_k$ such that all of the following conditions hold: $$ \text{ 1. } ...
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4answers
330 views

Monty Hall problem vs. roulette systems - how are they different?

So I got interested in the Monty Hall problem - I understand what it's about, but somehow I can't wrap my head around the idea of the final choice not being 50/50. More precisely: we all know (or ...
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2answers
1k views

Question about a program generating palindromic prime numbers

I'm a programmer and software designer. I'm definitely not a mathematician and my maths is quite basic. One of my colleagues challenged me to generate a palindromic prime number, at least 1000 digits ...
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2answers
162 views

Men on a boat problem

There is the usual question of some men on a boat- various men have various speeds, the boat has a capacity of 2 men, and the boat takes on the speed of the slowest man in the boat at any given time. ...
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1answer
100 views

Closest Packing of Spherical Caps

Let the surface $S_n$ of the unit ball in $\mathbb{R}^n$ centered at the origin $O$ be defined as the set of points $P(x_1,x_2,…,x_n )$ such that $x_1^2+x_2^2+⋯+x_n^2=1$. Let the spherical cap $C(α)$ ...
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1answer
140 views

Recreational mathematics - Digit sum

Find the sum of all 3-digit positive numbers N that satisfy the condition that the digit sum of N is 3 times the digit sum of N+3 Can you help me with this question?
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129 views

Card passing game, maximum length

Quoting from this question: There are 25 people sitting around a table and each person has two cards. One of the numbers 1,2,..., 25 is written on each card, and each number occurs on exactly two ...
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3answers
566 views

Problems like the handshake problem

I am in college and my RA has been putting up little thought problems on his door for us to see as we pass by, but the ones he puts up aren't too interesting. I wrote up the handshake problem (invite ...
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2answers
102 views

combinations problem about apples and pears

Carlo has six apples and six pears: how many ways he can set in a row 6 fruits so that there should never be a pear between two apples? Thanks in advance to everyone who will help me resolving this ...
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1answer
201 views

Do roots of a polynomial with coefficients from a Collatz sequence all fall in a disk of radius 1.5?

Consider a modified version of Collatz sequence: $C(n)=\left\{ \begin{array}{ll} \frac{3n+1}{2} & n\ \mathrm{odd} \\ \frac{n}{2}& n\ \mathrm{even}\end{array} \right.$ Let $F_n$ be the ...
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2answers
267 views

Vector spaces inquiry

Denote By $V$ the real vector spaces of all real polynomials in one variable, and let $P : V \rightarrow \mathbb{R}$ be a linear map. Suppose that $\forall$ $f,g \in V$ with $P(fg) = 0$ we have $P(f) ...
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2answers
295 views

A less challenging trivia problem

There are 25 people sitting around a table and each person has two cards. One of the numbers 1,2,..., 25 is written on each card, and each number occurs on exactly two cards. At a signal, each person ...