# Questions tagged [mathematics]

A puzzle related to mathematical facts and objects, whose solution needs mathematical arguments. General mathematics questions are off-topic but can be asked on Mathematics Stack Exchange.

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### It's what's within the two

What is the answer to this puzzle?
1k views

### Numbers that are averages of their digits

A number is called special if its decimal value^ is the average of its digits. How many special numbers are there? Bonus: How many special numbers do not begin with the digit 0? Good luck! ^ You are ...
2k views

### Reducing π to zero (again)

You are given the first 20 digits of π: 31415926535897932384. In each move, you can select a contiguous group of 5 digits and increase/decrease them all by the same integer, provided that each ...
2k views

### frog on a number line

A Frog is at C. The purple numbers are the probability that the frog jumps in that direction when it is at certain place. The frog stops at A or E. What is the probability that the frog stops at A? ...
2k views

### Interesting irrational number

Can you find an irrational number $x$ such that $x$, $1/x$ and $x^2$ all have exactly the same digits after the decimal point? Good luck!
1 vote
204 views

### Logic and Geometry Problem #5: does Savage Go have cycles?

My question is whether or not a cycle can occur in the game of Savage Go. That is, you kill some of mine, I kill some of yours, you kill some of mine... Endless cycle of turns. Game never finishes. No ...
1 vote
196 views

### How many 4x4 Latin Squares are there?

I thought of this problem when playing Sudoku. Let A = {1,2,3,4}. I have to make a 4x4 box (i.e. the size of A in both dimensions) and fill it with data such that ...
684 views

### Don't let 'em die!

There is an 8x8 array of sleeping humans with ten feet between adjacent humans, with sides in the four cardinal directions. A hero and a villain start ten feet west of the northwest human. The hero ...
271 views

### Waffleing my Egg

This morning I had a waffle and a fried egg for breakfast. The fried egg was cooked with a mold, so it was perfectly round and three inches in diameter. The waffle was a 3-inch by 4-inch grid of one ...
2k views

### Can you find a 3x3 white square somewhere in this relatively prime graph?

This puzzle comes from: http://skepticsplay.blogspot.com/search/label/puzzles Wow, it's been some time since I've posted a puzzle! Here's a simple pure math puzzle off the top of my head. Back in ...
310 views

### Permutations with given longest increasing subsequence

How many permutations of 1 to 20 are there with 2,5,6,9,13 as a longest increasing subsequence? (It may be tied with others.)
548 views

### Minimum K for detecting fake pearls in one weighing

There are 10 boxes, each with the same number of pearls (represented by K). Genuine pearls weigh 30g each, while fake pearls weigh 29g each. Each box is either all genuine or all fake, and any number ...
1 vote
78 views

### Can you find the missing letters?

A pattern of letters has had letters 10 through 20 deleted. Your job is to find out what letters should go in spots 10-20. Here is the pattern: a, b, c, d, d, d, f, e, f, _, _, _, _, _, _, _, _, _, _, ...
323 views

### Cutting the points evenly

I draw an even number of points on a piece of paper. Is it possible to cut the paper into two pieces with a single straight cut, such that: Each piece gets the same number of points The cut does not ...
183 views

### Puzzling Pelican Pebbles

Story Setup It's Percy the Pelican's first day running the front desk of his master's magical pebble shop. His job is to fetch pebbles from the stock room to fulfill customer orders. The stock room ...
1 vote
225 views

### #TenderfootSpiral

Clues: Two hikers on an incline arrow with numbers. Hiker 1 says: 38.548493. Hiker 2 says: -105.998749. Sequence: Hiker 17711 Hiker 0 377 0 144 1 10946 233 987 0 2584 6765 75025 Hiker = ...
223 views

### Intermingled primes

This puzzle is part of the Monthly Topic Challenge #14: Think inside the (very small) box!. 6 different, 3 digit primes are stacked here in two layers. You only see the sum of overlapping digits. ...
89 views

### Another paint balls problem [closed]

Suppose you have 2n balls with 1 of them being red and the rest being white. In each round the balls are randomly paired into n pairs. All white balls in a red-white pair are painted red. What is the ...
380 views

### Closed path on a dodecahedron

Your task is to draw lines between edges on a regular pentagon such that if you tile a dodecahedron with 12 identical copies of that pentagon you get a single closed line which does not intersect ...
197 views

### Expected number of steps [closed]

There are N cars that are parked in N parking spots numbered from 1 to N. After the Nth parking spot, there are N more parking spots numbered from N+1 to 2N. At each step, a car is selected randomly ...
135 views

### The Triangular Cannonball Problem [closed]

How many ways are there to stack an equilateral triangle of cannonballs into a tetrahedron of cannonballs? In other words, how many positive integers are both triangular and tetrahedral?
2k views

### A colorful dodecahedron

Divide a "base" edge of a regular pentagon into three equal parts. Then draw two lines from the base to the center of the other edges such that the lines do not intersect. This splits the ...
140k views

### Is this duplo train track under too much tension?

My kids were making this train track of duplo the other day, and this is what they put together. They are still very young, and for them, this is something big. They were really proud that they ...
207 views

### Figure Out The Language: Min

Imagine there is a programming language in which you can only evaluate expressions. Expressions have operators and constants. There are three types of constants. Numbers - Any number from 0 to ...
122 views

### Connect dots on a grid with one continuous line (optimization)

(This question is the third puzzle of the Connect dots puzzle series. You can find the first two puzzles here and here, respectively. The original question and photos originate from webadventurer. ...
188 views

### Find unique values for the variables to make all statements true

In this puzzle you have to find unique values for the variables to make all statements true. I have proceeded to here and now I seem to be stuck.
301 views

### Logic and geometry problem #4: are these games functionality equivalent?

Two games, Crossway and Mincut, are believed to be functionally equivalent. That is, a win by Crossway rules will necessarily lead to a win by Mincut rules. And a win in Mincut will be a win in ...
1 vote
129 views

### Regular polygons meeting at a point

How many ways can regular (convex) polygons meet at a point (vertex), so there are no gaps or overlaps? Here's an example with a square, hexagon, and dodecagon.
120 views

### Tile 1x2 dominoes in a 2x10 space

How many ways are there to tile unmarked 1x2 dominoes in a 2x10 space? Bonus: What if the dominoes were identical and had pips on their front (face-up), so they could be distinguished by 180 degree ...
161 views

### Find the Rat in the Hole

A rat is in one of 100 holes that are in a line. Each night he moves to the left or right 0,1, or 2 holes randomly selected. You pick one hole each day to look in until you find him. What procedure ...
750 views

### Poetry of the stars

Prof. Levenshtein is meant to be teaching us astronomy, but they fancy themselves as a poet. Can you figure out the name they've given to each star?
778 views

### Was Humpty Dumpty right?

Raymond Smullyan's What is the Name of This Book? contains a puzzle which I'll paraphrase here: Of the identical twins Tweedledum and Tweedledee, one of them lies on Mondays, Tuesdays and Wednesdays ...
700 views

### Number of 1's needed to write all primes up to P

i) Find, if it exists, a prime P such that the number of 1's used to write all the primes from 2 to P is precisely P. ii) Are there infinitely many such P? If not, find them all. These questions ...
534 views

### Game of the glasses on the windowsill

The windowsill above the sink is where my wife and I place our dirty wine glasses. And while both of us love each other, neither of us love loading the dishwasher. As a result, these dirty glasses ...
313 views

### Alice knows a+b, Bob knows a×b. Can we find (a,b)?

There are two distinct positive integers $a,b$ (i.e. $a\neq b$) such that $a < 7$ and $b < 7$. Alice has been told the sum of $a$ and $b$. Bob has been told the product of $a$ and $b$. Both ...
2k views

### 9 trees in 7 rows with 3 trees in each row

The following puzzle is a variant of a puzzle published in the May 8, 1926 issue of THE WINNIPEG TRIBUNE MAGAZINE: In the picture below there are nine trees arranged in two rows with five trees in ...
605 views

### Self-referential sequence that is sometimes powers of two

I've created an integer sequence where, after the first two elements, every element is calculated using the previous two. If the first two numbers are $1$ and $3$, the sequence goes as follows: 1, 3,...
159 views

### How to swap position of two elements when you can only rearrange four of them by one-way rotation?

The following puzzle is the final puzzle in the video game "Grim Tales: The Bride". Is there a methodical way of solving this type of puzzle? Can I plan steps to swap the position of the red ...
195 views

### Decipher a long sequence of numbers

51699624576811268526783109824167925197245652924177251787241782538786302873119768312852668312686625268245686752824473216878253830287925175273039778250852831287672527245665263006725397251762529312811498 ...
317 views

### Leonardo Da Vinci's Magic Calendar

It is the year 1468 and young Da Vinci has built a magic calendar going back to Jesus. Here's how it works: Given 7 distinct years between 33 and 1468 inclusive, the arranger discards one and orders ...
156 views

### Logic and geometry problem #3: are cycles possible in Scattercut with added "maximum" rule?

My question is whether or not cycles can occur in the game of Scattercut. That is, you kill some of mine, I kill some of yours, you kill some of mine... Endless cycle of turns. Game never finishes. ...
761 views

### What do 84, 96 and 108 have in common?

There's a certain property that's shared between (as far as I know) infinite positive integers including 84, 96 and 108. Below are the first thousand numbers with this property; I added that many in ...
73 views

### Find the largest number with unique digits, that is divisible by each of its digits [duplicate]

Find the largest number with unique digits, that is divisible by each of its digits. For example 132 has unique digits and is divisible by 1, 2 and 3. But it is not the largest such number. Can you ...
197 views

### 14 coins problem but you can't understand the scale

The 12 coins 12 coins problem but you can't understand the scale asks for is not the maximum possible, therefor this follow-up question: You have a number of coins, one is fake but you don't know ...
216 views

### How many solutions to the twelve coins problem are there?

I was recently asked to solve the classic twelve coins balance problem: You are given twelve coins, eleven of which are equal in weight and one of which is either heavier or lighter than the other ...
635 views

### Pay each amount with at most two coins

Euro cent coins come in the denominations 1, 2, 5, 10, 20 and 50 cents. You are inconvenienced by the fact that you need a lot of coins to pay each amount up to 100 cents. To pay 99 cents, you need 6 ...
6k views

### 2 vs. 1.005^200

Without using a calculator or a computer can you determine which of these two numbers is bigger: $2$ or $1.005^{200}$ ? I saw this puzzle in a YouTube video, which I will post later.
490 views

### Colliding Bullets again

Here's a colliding bullets problem of my own devising that's different from previous versions. Every second a gun has a 60% chance of firing a bullet in a straight line. After 10 seconds there would ...
1 vote
Some malevolent entity (me) asks you to construct a Turing Machine which, given an input on its tape of the form $LbR$ where $b$ is some binary string, changes this to $Lb^{-1}R$ then halts (where \$b^{...