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

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13
votes
2answers
134 views

Six of a kind .

$$\begin{align} 2^{\color{pink}0}+7^{\color{pink}0}+8^{\color{pink}0}+18^{\color{pink}0}+19^{\color{pink}0}+24^{\color{pink}0}&=3^{\color{pink}0}+4^{\color{pink}0}+12^{\color{pink}0}+14^{\color{...
0
votes
1answer
61 views

How do you compare carsharing plans to calculate the cheapest?

Call hourly rate = HR. Assume that I can guess my monthly usage in hours, which I call $g$. Beware that the fixed fees are presented in different units of time, so first convert everything into months,...
0
votes
1answer
33 views

A prove for information restoration with 2 schedules that delete information

What kind of mathematics or technique do I need to use the following? Just pointing me in the right direction is also helpful as I love mathematics but I am not so good at it. It's a problem I have ...
22
votes
7answers
4k views

A riddle for 2015

How can one get $2015$ using $1,2,\dots,9$ in this order and only once, with the operations $+,-,\times,/$ ? Solving this riddle with a computer (using python) turned out to be impossible for me due ...
12
votes
1answer
483 views

New Year Combinatorics 2015

In the spirit of the festive period and in appreciation of the encouraging response to my X'mas Combinatorics problem posted recently, here's one for the New Year! Express the following as a ...
0
votes
3answers
77 views

There are two cars…

Let's imagine a $6000 km$ stretch of road. Now, there are two cars $A$ and $B$, each with average speeds of $100km/h$ (for $A$) and $250km/h$ (for $B$) respectively. If $A$ is given a headstart of $...
37
votes
1answer
592 views

Does my “Prime Factor Look-and-Say” sequence always end?

I'm trying to create a challenge for PP&CG where the object will be to find the longest sequence in a given time, but I'm worried that there may be an infinite sequence that will ruin things. The ...
1
vote
1answer
963 views

Possible mathematical finishes to the darts game 501

I was recently posed a question by a friend - How many possible finishes exist within the darts game 501 which include 3 (or more doubles) and using no more than 9 darts? For those unfamiliar ...
1
vote
1answer
125 views

Is there a proof that zero multiplied by infinity = a real number [duplicate]

Someone told me that $0\times \infty = 1$. I am baffled by this because I thought you cannot multiply by infinity because it isn't a real number. If you can, is it possible to explain how and give ...
1
vote
1answer
137 views

Why does strategy-stealing not work for Go?

The related Wikipedia article states: In Go passing is allowed. When the starting position is symmetrical (empty board, neither player has any points), this means that the first player could steal ...
31
votes
1answer
1k views

Xmas Combinatorics 2014

Inspired the various** algebraic X'mas greetings sent to me over the festive period, I thought I would try to devise one of my own. $$\Large \color{red}{\sum_{i=a-1}^{r-1}}\color{green}{\sum_{j=s-1}^...
36
votes
6answers
4k views

Function whose third derivative is itself.

I'm looking for a function $f$, whose third derivative is $f$ itself, while the first derivative isn't. Is there any such function? Which one(s)? If not, how can we prove that there is none? Notes: ...
4
votes
1answer
107 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 $n=1,...,10$. ...
-9
votes
2answers
4k views

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

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 $. ...
3
votes
1answer
99 views

Puzzling Sequence [closed]

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
54 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 cross-...
1
vote
0answers
68 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
102 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
53 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 $k$...
0
votes
0answers
94 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
585 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
57 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 : ...
1
vote
1answer
1k 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
172 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 $S(S(n))=2$
3
votes
2answers
112 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
67 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
117 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
127 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 ...
2
votes
2answers
115 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 ...
-1
votes
1answer
74 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
4k 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
107 views

Cutting chocolate diagonally

Given is chocolate with rectangular pieces of size $a \times b$. If cut diagonally, how many pieces will it be split into? If knife passes exactly by co-catenating we assume there is no damage to ...
7
votes
1answer
170 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
34 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
180 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
64 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
70 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
72 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
127 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
337 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 ...
2
votes
1answer
168 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. ...
18
votes
1answer
465 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
284 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, \dots\}...
7
votes
4answers
193 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 $x, y$ ...
6
votes
2answers
178 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
91 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
40 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 ...
26
votes
5answers
13k 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 1....
0
votes
1answer
73 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 ...