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

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0
votes
1answer
121 views

Spirit of the holidays. [closed]

My teacher gave us this homework problem to do over the break, and said the solution should be incredibly obvious, but I can't seem to figure out where I should start...can anyone help?! ...
4
votes
1answer
87 views

Analytic solutions to a simple math trick

As proven here $3816547290$ is the only positive integer in which every digit is used; each digit is used only once; the first $n$ digits are divisible by $n$, for $n=1,...,10$. ...
-7
votes
1answer
113 views

Why is it possible to find the birth year by subtracting one's age from 114?

I noticed that any person can find their birth year just by subtracting their age from the number $114$. For example, if I am $25$ years old then from $114-25=89$ I know the birth year is $1989 $. ...
4
votes
1answer
57 views

Puzzling Sequence

Today I was given a question that first I thought might be easy to solve but then no matter how hard I tried I couldn't solve it.(It's not really related to maths just some puzzle) if: $$ 9999=4\\ ...
1
vote
1answer
42 views

How to calculate 2-d plane from 3 4-d points?

I want to compute 3-d cross-sections of a pentatope (4-dimensional tetrahedron). The 3-d cross-sections will be calculated as: x+y+z+w=c C is a constant that I will vary to get different ...
1
vote
0answers
51 views

Where can Gaussian Elimination be used?

I have searched for this and came to know about it that it is traditionally used to solve linear equations, finding determinant, rank of matrix, inverse of matrix. There was a problem on codechef: ...
1
vote
1answer
75 views

subtle/annoying fallacious proofs [duplicate]

I've been invited to a maths themed Xmas after party. I need to prepare a selection of interesting, and relatively simple fallacious proofs which other guests will try and find the flaw in. I'm trying ...
1
vote
1answer
33 views

Solving a reaction-diffusion problem using Separation of Variables

$$U_{t} - D U_{xx}= -kU$$ where BC: $U_{x}(0,t)=0$, $U_{x}(l,t)=0$ where $0 < x < l$, $t > 0$ IC: $U(x,0)=A + B cos \big(\frac{2πx}{l}\big)$ where $ 0<x<l$ where $D$ and ...
0
votes
0answers
73 views

Maths to take a user chosen number to a predictable number

As part of simple card trick, I want to allow a user to choose a number between 1 and 100 and then ask them to do various maths to lead them to the same number so their choice becomes irrelevant. One ...
15
votes
1answer
562 views

The 'Unlock All Digits' Game

I challenged myself and thought of a new problem I tried to solve. Here are the rules : The goal is to 'unlock' all the numbers $0,1,2,3,4,5,6,7,8$ and $9$ When you start the game, the only number ...
2
votes
3answers
50 views

$n$ is twice the sum of squares of digits of $n$

Let $f(n)$ denote the sum of squares of digits of $n$, that is $$ f(10k+r) = \begin{cases} r^2 + f(k) &\text{for }10k+r \neq 0,\\ 0&\text{otherwise}. \end{cases} $$ I've found (while ...
15
votes
4answers
2k views

Solving 9 sons puzzle

The following math puzzle : ...
2
votes
1answer
33 views

Special Binary Relations/ Empty Relation, Universal Relation And identity Relation?

The universal relation U = A × A. (Correct me if I'm Wrong). I believe that the Universal Relation is an Equivalence Relation The empty relation E = ∅. From my understanding, a Empty relation on a non ...
3
votes
5answers
89 views

Find the number of all 3 digit numbers $n$ such that $S(S(n))=2$

For any natural number $n$ ,let $S(n)$ denote the sum of the digits of $n$.Find the number of all 3 digit numbers $n$ such that $S(S(n))=2$
2
votes
2answers
75 views

Santa is secretly deranged! or, how to hand-generate assignments for a gift exchange?

Consider a standard Secret Santa/gift exchange game draw. We have a pool of $n$ people, each of whom is supposed to be assigned another member of the pool to find a gift for, without the recipient ...
0
votes
1answer
52 views

Multiply large numbers

Consider the product $723145878987 \times599987871$. If I want to know that what would be sum of unit and tens digit of the result then Is there a trick that I could find it as fastly as possible?
3
votes
1answer
80 views

Can the wolves catch the hare?

Say you have 7 positions. 1 Hare and two Wolves in the following starting positions:    H o     o W   W  o   o The hare can take a step of size 2. The ...
1
vote
2answers
99 views

switch the colour until only one black square is left

Consider a standard chess board (8 × 8 squares). In each move, you pick one row or one column and switch the colours of all 8 squares (from black to white or from white to black). Is it possible to do ...
1
vote
2answers
86 views

How can the sniffer dog find the bag of drugs?

There are $n$ bags. In one of the bags are drugs. There is a dog that when given a group of bags, can tell whether there are drugs in the group or not. Each sniff counts as a "turn". What is the best ...
0
votes
1answer
44 views

Geometric Interpretation of Trigonometric Ratios

Is there a "good" geometric interpretation of trigonometric ratios for complex values? For example, we know that $$\cos(z)=\frac{e^{iz} + e^{-iz}}{2}$$ for all complex $z$ but is there a way to ...
0
votes
0answers
113 views

Calculating equal playing time in a soccer game with minimum number of changes.

I need to produce a formula that takes the following parameters: T = time of game in minutes p = number of players on field at one time s = number of substitute players Each of these is variable ...
1
vote
1answer
77 views

Cutting chocolate diagonally

Given is chocolate with rectangular pieces of size $a \times b$. If it will be cut diagonally, how many pieces will be splitted? If knife pass exactly by concatenating we assume there is no damage ...
0
votes
0answers
27 views

finding angle of spirals along with length of it's line at a certain point

I'm tying to calculate the angles (the angle between each line segment and a horizontal ray to the right blue) of spirals at a certain point along with figuring out the length of the other lines. see ...
7
votes
1answer
151 views

How Do I Find My Car

I have been discussing this problem with a coworker for a few days now and neither of us have made any headway on it. I would appreciate any help with a possible solution or maybe a suggestion of a ...
0
votes
0answers
31 views

$n$th number of concatenating consecutive integers [duplicate]

How do I find the nth digit of concatenating consecutive integers as in: $123456789101112131415161718\cdots$ where the $10th$ digit = 1$ , $11$th$ = 0$, $12$th $= 1$, $13$th $= 1$ $\cdots$ How do I ...
0
votes
1answer
48 views

Conditions for magic square.

So I've messing around with magic squares and something occured to me: Assume we have a nxn grid of numbers which respects the sum conditions of a magic square as in it has the appropriate column, ...
0
votes
2answers
38 views

Recreational chess questions based on the knights

I basically know whether the following statements are true, but I would like to know how they are proved. A knight kept anywhere on an empty chess board can not reach its adjacent square in exactly ...
0
votes
3answers
66 views

An recreational question on analysis

Alice and Bob ran a marathon ($26.2$ miles) with Alice running at a uniform $8$ minutes per mile pace and Bob running erratically, but taking exactly $8$ minutes and $1$ second to complete each mile ...
6
votes
1answer
69 views

Inequality: $(a^3+3b^2+5)(b^3+3c^2+5)(c^3+3a^2+5) \ge 27(a+b+c)^3$

Proving inequality for positive real $a,b,c > 0$: $$ (a^3+3b^2+5)(b^3+3c^2+5)(c^3+3a^2+5) \ge 27(a+b+c)^3$$
1
vote
3answers
89 views

Olympic elementary combinatorics problem

This is a problem taken from the regional selections of the Italian mathematical olympiads: A knight is placed on the bottom left corner of a $ 3\times3 $ chess board. In how many ways can you move ...
6
votes
3answers
173 views

Secret Santa Perfect Loop problem

(n) people put their name in a hat. Each person picks a name out of the hat to buy a gift for. If a person picks out themselves they put the name back into the hat. If the last person can only ...
0
votes
1answer
44 views

Basic examples of probabilistic method

I'm looking for a truly basic example of probabilistic method proof which could be presented without a board (i.e. speaking only), that is, even moderately complicated calculations are not allowed. ...
13
votes
1answer
260 views

Infinite prisoners with hats — is choice really needed?

The problem is this (recently asked about here): A countably infinite number of prisoners, each with an unknown and randomly assigned red or blue hat line up single file line. Each prisoner faces ...
5
votes
2answers
154 views

Prisoners Problem

We have an infinite number of prisoners enumerated $\{1, 2, \dots\}$, and on each prisoner there is a hat of either blue or red color. The $n$th prisoner sees the hats of prisoners $\{n+1, n+2, ...
7
votes
4answers
176 views

If $(x+\sqrt{x^2 + 1})(y+\sqrt{y^2 + 1})=p$, find $x+y$

I was given this factorization problem and I tried many things, but couldn't solve it. Can someone, please, give me a hint? If $(x+\sqrt{x^2 + 1})(y+\sqrt{y^2 + 1})=p$, find $x+y$. Here $x, y$ ...
13
votes
0answers
201 views

Sheldon Cooper Primes

On the $73^{\text{rd}}$ episode of the Big Bang Theory, Dr. Sheldon Cooper, an astrophysicist portrayed by Jim Parsons $(1973 - \stackrel{\text{hopefully}}{2073})$ revealed his favorite number to be ...
6
votes
2answers
161 views

What is the coefficient of $x^{25}$ in $(x^3 + x + 1)^{10}$?

Here's what I have so far on the off chance that my thinking is correct... So using Vieta's the coefficient of the $x^{25}$ should be $-(r_1r_2r_3r_4r_5+r_1r_3r_4r_5r_6+...+r_6r_7r_8r_9r_{10})$ Since ...
1
vote
3answers
57 views

Combinatorial card game [duplicate]

There is a card game I've played before, where it goes as follows: You take a standard deck of cards, and shuffle them randomly. You then proceed by flipping each card and placing them down, ...
1
vote
0answers
26 views

Adding a factor to a ranking?

I have a ranking of 10 items from best to worst. Let's assume that the best is ranked 1 and the worst is ranked 10. Each item is ranked according to some rules that we cannot know so all we get is the ...
0
votes
0answers
25 views

making /adjusting a perodic signal given an equation

I know if I have the sin wave equation Asin(2pif*t+phase) I can increase/decrease/adjust the periodic frequency of the signal by changing f. But if I have the following equation below how can I also ...
19
votes
5answers
1k views

Puzzle of gold coins in the bag

At the end of Probability class, our professor gave us the following puzzle: There are 100 bags each with 100 coins, but only one of these bags has gold coins in it. The gold coin has weight of ...
0
votes
1answer
64 views

sequence get number in sequence from place in sequence

There is a sequence $$X = {1,1,1,1,1,1,1 \dots 2,2,2,2,2,2,2,2 \dots,3,3,3,3,3,3 \dots 4,4,4,4,4,4,4 \dots (k-1),(k-1),k}$$ So there are $(k)$ 1's, $(k-1)$ 2's and $(k-2)$ 3's and so on. Is there ...
1
vote
1answer
45 views

Tree recursive question: number of nodes and relationship with children

Suppose a given tree T has n1 nodes that have 1 child, n2 nodes that have 2 children, . . . , nm nodes that have m children and no node has more than m children, how many nodes have NO child are there ...
2
votes
3answers
64 views

Fun problem. Apparently $\prod_i(1-p_i) \geq 1 - \sum_ip_i$ with $p_i \in [0,1]$ is always true. But how to demonstrate it?

so, I want to demonstrate the validity of the following inequality: $$ \prod_i(1-p_i) \geq 1 - \sum_ip_i $$ with $p_i \in [0,1]$, it is always true, which it seems to be always the case if you test ...
3
votes
2answers
166 views

how to solve triangles count puzzle

Below is a puzzle of counting triangles.How to solve such puzzle ? source: http://gpuzzles.com/mind-teasers/how-many-triangles-challenge/?source=stackmath
0
votes
1answer
50 views

Which Snake fields can be played infinitely long?

Snake is a very old game for phones. Its a 'real time game', that means you have to make decisions fast. The rules are: You are a snake. You can move to the left, to the right or go straight ahead. ...
2
votes
1answer
44 views

How big can the deck get while still allowing this puzzle to be solvable?

Here's a classic puzzle (I think Martin Gardner talks about it somewhere, though I'm not sure exactly where): Alice and Bob are co-conspirators. Alice is dealt five random cards from a standard ...
0
votes
2answers
59 views

Combinatorics type

I have this problem: From a set of numbers, such as $\{1,2,3,4,5,6\}$, a new set is created containing all the possible single pairs. ie. $\{12,13,14,15,16,\ldots\}$. Another set contains all the ...
0
votes
1answer
56 views

Let's say you're playing ping pong. For each point you win or lose, how would you update your probability of winning the next point?

Let's say I start out believing that my probability of winning the next point is in the interval $[0.25, 0.5]$ with 50% confidence. If I win the next point, what is an intuitive or "good" way to ...
0
votes
2answers
47 views

Birthday problem, the hard way(not using 1-unfavourable outcomes).

How would you go about calculating the chance of two people having the same birthday in a room of 3 people and a year consisting of 365 days?