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

learn more… | top users | synonyms (2)

13
votes
5answers
3k views

Can you complete the expression 2 _ _ _ _ 5 = 2015?

Can you complete the expression 2 _ _ _ _ 5 = 2015 and make it correct by replacing two underscores with a selection of the operational symbols $+, - , ...
1
vote
1answer
32 views

Custom Weighted Formula

I'm in need of a mathematical formula that will be ultimately utilized in any programming language that would give me a value that I could ultimately sort or rank by. I have 2 variables: variable 1 = ...
0
votes
0answers
31 views

Find number of element in $\{m\in\mathbb N:m\leq n\text{ and }m\text{ has the digit 3}\}$.

Inspired by a youtube video claiming that "almost all positive integer has the digit 3", I set myself a challenge: Give a formula, in terms of $n$, for the number of positive integer that is less ...
1
vote
0answers
64 views

Prove Pythagoras' Theorem by using the area of a kite [on hold]

Given a kite which can be dissected into two right triangles, use the area of the kite to construct a proof of Pythagoras' Theorem for a triangle with one side double the length of another, preferably ...
1
vote
1answer
76 views

Finding $a^{2014} + b^{2014} + c^{2014}$ given some conditions on $a,b,c$.

I came across this problem: "Let $a$, $b$, $c$ be nonzero real numbers that satisfy the conditions : $$a + b + c = 9,\\\mathrm{and}~ab + bc + ca = 27 $$ Calculate $$a^{2014} + b^{2014} + ...
-1
votes
2answers
35 views

Find the missing number in the given series? [on hold]

Find the missing number in the given series? the series is $1584$, $900$, ____, $180$, $48$, $4$.
1
vote
0answers
57 views

Is it possible to solve sudoku without backtracking?

I occasionally solve sudoku puzzles on smartphone in spare time. My approach is based on the belief that at each stage in solving a sudoku puzzle there is at least one cell where there in only one ...
1
vote
1answer
28 views

Probability of 2 students being chosen the both have under 100 books at home

Suppose we select two students at random from the class of fifteen. What is the probability that both students chosen have less then 100 books at home? Data provided is the amount of books each ...
2
votes
2answers
88 views

Digit sum of $n^2$ is 44

Is there a whole number $x$ such that the sum of the digits of $x^2$ equals 44? I would like someone to tell me if my thoughts are correct. The remainder of a number a divided with 9 is the same as ...
1
vote
2answers
75 views

The product of two prime numbers

I have two expressions (both of which have a term raised to the power of $n$) and I am trying to prove that they can't be prime numbers at the same time for $n>2$. I can't post the expressions, ...
0
votes
3answers
52 views

Is there a pattern present?

Jack is looking at Anne, however Anne is looking at George. Jack is married, George isn't, and Anne's status is unknown. provided this info alone determine whether: a) A married person looks at a non ...
0
votes
0answers
27 views

Thermodynamics based proofs

What are some mathematical inequalities and theorems that follow using thermodynamics "proofs" (rigorous or just intuitive)? Any suggested books on the matter? For example, AM-GM inequality follows ...
0
votes
1answer
50 views

A vessel contains $x$ amount of milk out of which $y$ amount is taken out and replaced with water $n$ times.

There is a formula in my book for questions of type, A vessel contains $x$ amount of milk out of which $y$ amount is taken out and replaced with water. After $n$ such operations what will be the ...
1
vote
1answer
32 views

How to solve problems on alligation and mixture when three types are given?

Suppose there are three qualities of rice, A(1 dollar per Kg), b(2 dollar per Kg) and C(3 dollar per Kg). The salesmen want to mix these in a certain ratio a:b:c so as to make the price 2.5 dollar per ...
1
vote
2answers
58 views

Building a box from smaller boxes

John has 77 boxes each having dimensions 3x3x1. Is it possible for John to build one big box with dimensions 7x9x11? I'm leaning towards no, but I would like others opinion.
0
votes
1answer
49 views

Maximum number of chess moves

Chess has a limited number of maximum moves because of the 50-move rule (50 moves without any captures or pawn moves results in a draw). There are 30 capture-able pieces, and I've figured out that the ...
0
votes
1answer
31 views

Lottery probability with payout system

Assume we have a lottery which has following payouts 1,2,5,6,9,10,16. The organizer expects 4% profit from the lottery. I wrote ...
11
votes
0answers
97 views

Penrose's remark on impossible figures

I'd like to think that I understand symmetry groups. I know what the elements of a symmetry group are - they are transformations that preserve an object or its relevant features - and I know what the ...
4
votes
0answers
34 views

How many ways I can put $k$ bishops on $n\times n$ chessboard?

Is there a formula how to count in how many ways I can put $k$ bishops on $n\times n$ chessboard such that no two bishops threaten each other?
1
vote
2answers
54 views

Ten marbles put in a box, colour of each set by toss of a fair coin. You draw (with replacement) ten white marbles. Probability all marbles are white?

The following question comes from the probability section of the Titan Test*. * I will avoid the debate around whether this test accurately measures what it aims to, nor whether such aims are ...
10
votes
1answer
74 views

A generalisation of Napoleon's theorem. Is this result original?

I've found a generalisation of Napoleon's theorem to general polygons. Take any regular $n$-gon inscribed in a circle and stretch it (in any direction) so that the circle becomes an ellipse and the ...
3
votes
1answer
67 views

The jackal, the lion, the parrot and the giraffe - logic puzzle

Here is a puzzle that appeared in a Russian magazine named Kvantik (see Tanya Khovanova's Math Blog). [The trick lies in that we don't know exactly what the hedgehog knows at each stage. The symbology ...
3
votes
1answer
56 views

find a group of lowest N numbers so that no 2 pairs have the same bitwise or

I am trying to find the lowest group of N numbers (i.e. N=1000) so that no 2 pairs from the group have the same bit-wise or. more specific need to find a group $A = \{a_1,a_2,a_3,..,a_N\} $ such ...
2
votes
2answers
52 views

Sailor,Monkey,Coconut answer in elaborate

In Sailor, Monkey, Coconut Problem Can anyone tell me how adding 56 gives me another solution??I understand that cocount is divided into 5 piles.But how is 56 give me another solution?why wouldn't ...
4
votes
3answers
54 views

An interesting wire-tying problem to match wire ends in as few trips as possible

You have $N$ wires that all extend from one location to a second distant location. The wire ends at both locations are unlabeled, and the goal is to label them all (on both ends) with distinct labels ...
1
vote
0answers
47 views

Proof Attempt of Brouwer (via Separating Hyperplane Theorem)

In part motivated by the discussion here, I have been playing with trying to prove Brouwer's theorem appealing as minimally as possible to topology. In the 1-dimensional case I believe one can ...
1
vote
0answers
36 views

A harder long division puzzle than the first; what should “Algebra I” solution look like?

Here's another problem, significantly harder than the first, but still accessible to target audience. The statement of the problem (i.e., northwest corner only) comes from a PennyDell puzzle magazine: ...
4
votes
1answer
154 views

Cover the grid graph with simple cycles

I have a two dimensional n x m grid graph. And I want to find in how many ways this grid can be covered with simple cycles (it can be a one cycle or it can be many ...
6
votes
3answers
97 views

“Long-division puzzles” can help middle-grade-level students become actual problem solvers, but what should solution look like?

This is my first post. I hope it's acceptable. EDIT Since there are people to whom such notation is foreign, I will point out that the problem represents KRRAEE / KMS, where PEI is the quotient and ...
2
votes
0answers
20 views

The odyssey of spies: Kryptos

The part four $K4$ of Sanborn sculpture, a sculpture located on the grounds of the CIA in Langley remains unsolved. As you can read in [1], Sanborn released a clue for the 64th-69th letters in part ...
2
votes
0answers
49 views

Beal Conjecture and ($\bmod 3$) operation [closed]

When we apply a ($\bmod 3$) operation on the $A^x +B^y =C^z$ we will see some strange results. For e.g.: Let $A=6m+1$ & $B=6n+1$. Since $A$ & $B$ are odd numbers, $C$ will have to be even. ...
2
votes
1answer
39 views

Code Jam 2014 Cookie Clicker Alpha Proof

I was looking at the solution for the Code Jam 2014 qualification question but the proof of correctness seems to be incomplete and I was wondering if anyone could help me with it. The full question ...
9
votes
1answer
73 views

Branching Paths Problem

I was drawing some shapes during class, and I came across the following problem. If one takes steps of constant length, and one must deviate a constant angle $\alpha$ from one's previous step either ...
-1
votes
0answers
27 views

sum percentages without knowing totals

lets say that for "business choice" I have in my DB, for each item in my list, just percentage of overall sold items but not the count of them. and in another file just what i sold today so DAY1 ...
1
vote
1answer
37 views

How to get the interest rate per quarter given the semiannual interest rate?

I have this economics problem. What is the present worth of $500.00$ Rupiah deposited at the end of every three months for $6$ years if the interest rate is $12\%$ compounded semiannually? ...
5
votes
1answer
140 views

Stupid numbers game

Occasionally I encounter a recurring instance of stupidity when I teach mathematics, namely that someone asks if we can play a game known in Danish under the name "bum", since that person thinks that ...
7
votes
1answer
96 views

Smallest cylinder into which a regular tetrahedron can fit?

Given a regular tetrahedron (as shown) of edge length $b$, determine the diameter $d$ of the smallest right circular cylinder (pipe) of infinite length along which the tetrahedron can slide.
3
votes
1answer
67 views

Period of Fibonacci mod $b$?

It is not too difficult to show that the Fibonacci numbers mod $b$ form a periodic sequence. I would like to say something interesting about the period. There is a small shortcut to the brute-force ...
0
votes
1answer
32 views

Upper bound of digit sum of powers

Take $x \in \Bbb N$, $x \le9$ and $m \in \Bbb N$. Now we define a function $d_s(n): \Bbb N \to \Bbb N$ as the digit sum of $n$ in base $10$. Now let's say we have a lower bound $b_l$ and an upper ...
0
votes
0answers
26 views

recursion $t(n)=\sqrt{2} \times \frac{tn}{2} +\log{n}$

I tried substituting $m=\log{n}$ $t(2^n)=\sqrt{2} \times \frac{t2^n}{2}+m t(m)= \sqrt(2) \times \frac{tm}{2}+m$ From here I got $\log {n}$ But with induction I proofed its $\sqrt {n}$
2
votes
0answers
44 views

Why 6 races is not sufficient in the 25 horses, 5 tracks problem

The horse-racing puzzle has been asked on mathSE several times (1, 2, 3, 4); there is also a generalization. I restate the puzzle below: 25 horses all run at different speeds. You can race 5 ...
1
vote
4answers
39 views

If $n$ people are placed in a room, prove that at least $2$ of those people will have the same number of friends in the room.

If $n$ people are placed in a room, then at least $2$ of those people will have the same number of friends in the room. I want to prove this statement. Here are some of my thoughts: If all the ...
2
votes
1answer
93 views

Most Number Of Lowest Numbers

I'm looking for a formula for the following problem. Hopefully I can explain this clearly and it all makes sense. No, it's not my homework, it's part of a competition I'm involved with managing and ...
5
votes
2answers
344 views

Impossible numbers drawn from tricky function

The function is this: $f(\frac{a}{b},\frac{c}{d})=\frac{a+c}{b+d}$ where $0\lt \frac{a}{b} \lt 1$ $0\lt \frac{c}{d}\lt 1$ $a,b,c,d$ are all integers $a/b$ and $c/d$ are in lowest terms Are there ...
17
votes
4answers
408 views

Why do siamese magic squares have real eigenvalues, symmetric around zero?

There is a standard method to construct magic squares of odd size, known as the Siamese construction. I'll write $S_m$ for the $m \times m$ Siamese square. For example, here is $S_5$. ...
0
votes
1answer
73 views

Evaluating function

Here is the function: $$f(a)=\sqrt{f(a)+\sqrt{f(f(a))+\sqrt{f(f(f(a)))+\cdots}}}$$ Is there another way to represent this function so that it only has $f(a)$ on one side and no $f(a)$'s on the other ...
0
votes
2answers
39 views

Eliminating non-integer solutions to $ab / (2\sqrt{ab} + a + b)$

I am writing a program to output all $a,b \in \mathbb{N}$, where $a \le b \le n$ (for a given $n \in \mathbb{N}$), such that $$ \frac{ab}{2\sqrt{ab}+a+b}=c\in \mathbb{N} $$ For example, $a=9$, ...
0
votes
0answers
16 views

Lines of intersection of four similar planes are coplaner.

If lines joining four corresponding vertices of two tetrahedrons are concurrent, then the lines of intersection of four corresponding planes are coplanar, and the converse also holds true. What would ...
-4
votes
1answer
55 views

How many minimum weights do you need to measure all weights from $1$kg to $1000$kg [closed]

You can place weights on both side of weighing balance and you need to measure all weights between $1$ and $1000$. For example if you have weights $1$ and $3$, now you can measure $1,3$ and $4$ like ...
33
votes
3answers
924 views

Guessing the length of a playlist on “shuffle random?”

The other night I was hanging out with some friends and someone put on a playlist on shuffle random, where the songs are drawn uniformly at random from a fixed playlist. The person who put the ...