# Questions tagged [puzzle]

For questions about the mathematical principles behind puzzle, games, riddles, or their possible solutions. Questions that are not strictly mathematical in nature should be asked on Puzzling Stack Exchange.

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### Identify $d$ heavy coins where $d$ is unknown.

You are given $N$ coins which look identical (assume $N = 2^k$). But actually some of them are pure gold coins (hence are heavy) and the rest are aluminum coins with thin gold plating (light). You are ...
34 views

### How many locks and keys: combinatorics problem [duplicate]

A village keep all their most precious belongings in a vault. The vault has a certain number of locks, each lock with an individual and specific key. The people in the village want to make sure that ...
1 vote
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### Expectation of removing subtree

You are given a rooted tree with n nodes. On each step, you randomly choose a node and remove the subtree rooted by that node and the node itself; until all of them have been removed (that is, the ...
1 vote
75 views

### The best strategy that the fireman should use in this problem is?

Problem:Suppose there is a building with $10$ floors. Each floor has only one room and each room has exactly one person in it. On the first floor there is a child in the room who weighs $10$kg, on the ...
93 views

### Can we generate a valid $9\times 9$ sudoku using this algorithm?

Begin with a board of $9*9$ cells, each of the cells has no value but is possible to contain a number from $1$ to $9$ (I will call the numbers can be assigned to a cell is guesses; the amount of ...
1 vote
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### Expectation problem of postman

A postman brought N letters to a house with two letter-boxes. Since the two boxes were empty, he puts 1 mail in each of the two mail boxes. Then he chooses one of boxes with probability proportional ...
80 views

### Can anybody tell me how to find the value of $\alpha$?

I don't know how to find the value of $\alpha$: $$(\sqrt{3}+\sqrt{2})^\alpha + (\sqrt{3}+\sqrt{2})^\alpha = 10$$ I tried to simplify it to $$(\sqrt{3}+\sqrt{2})^\alpha=5$$ then take the ($\ln$) of ...
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### Expected number of children until more girls than boys

Consider a couple that has decided to have children. They will continue having children until they have more girls than boys. What is the expected number of children they will have?
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### Puzzle of an ant rearranging stacks of seeds in a line [duplicate]

Interesting puzzle that I haven't been able to solve or find a solution to. An ant rearranges a line of stacks of seeds as follows: With each iteration, the ant goes to each stack in order and grabs ...
1 vote
106 views

### Infected Dinner Brainteaser

I came across this brainteaser online that I found quite confusing: There are $1000$ people having dinner at a grand hall. One of them is known to be sick, while the other $999$ are healthy. Each ...
110 views

### A Simpsons Riddle [closed]

This riddle (from Amann and Escher) has been bothering me more than it should. What is going on here? I'm not sure I see how to translate this into the sort of propositional logic I can manipulate via ...
128 views

### Check my solution on a card flipping game

I'm trying to solve puzzle. Can someone please check my solution? Thank you! I have four cards with values $1,2,3,4$ faced down. Each turn, I can either flip a card or do nothing. If I flip a card, I ...
90 views

### Defective clock questions - Watch gaining 3 minutes every hour

An analog watch gains 3 minutes every hour. If it is set right at 11 a.m. on February 21st, 2012 when will the hour hand of this defective watch and a correct watch be at the same position ? My ...
373 views

### Impossible? Finding a closed loop on a six-sided die that crosses face $n$ exactly $n$ times, without returning to a face from one step before

I'm new here, so please tell me if I'm doing anything wrong. Here is a puzzle that I came up with, which I believe falls someone in the realm of graph theory or topology (I could be very wrong, please ...
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### Peter winkler's "Numbers" puzzle "Zeroes, Ones, and Twos"

I have a problem with the solution for the (b) part of the problem. The problem is as follows: Let $n$ be a natural number. Prove that $2^n$ has a multiple whose representation contains only ones and ...
1 vote
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### Family Members Birthday Dates all different, but our birthdays will fall on same day, even Leap Years. There is a total of 9 in this Birthday Club. .

I can compile a list if needed and post, but I noticed this over 50 years ago, My Father, My Brother and Myself our Birthdays fall on the same day of the week every year. Even Leap years, that does ...
1 vote
65 views

### Using tricks to Maximize a Product of Binomial Coefficients $\binom{n}{x} \times \binom{n}{b - ax}$

I'm trying to find a solution to the following problem without resorting to brute force: \begin{equation} \text{maximize } \binom{n}{x} \times \binom{n}{y} \end{equation} subject to the constraints: \...
99 views

### Double the marbles in bucket to produce empty bucket [closed]

Three buckets have marbles. You are allowed to double the number of marbles in a bucket by borrowing from one of the other two buckets. Prove that it is possible to produce an empty bucket with a ...
204 views

### Barge Problem from 26 Years of Posing Problems [closed]

I'm struggling to find the solution this problem: A boy at the stern of a canal barge leaps off onto the tow path and while the barge keeps moving, runs along the path until he gets to the bow, where ...
62 views

### Dropping eggs from a building with unknown floors

With two eggs and a building with 100 floors, what is the optimal strategy for finding the lowest floor at which an egg will break when dropped? Followup: What if the number of floors of the building ...
31 views

### Polynomial guessing with 2 queries [duplicate]

Alice is allowed to choose an arbitrary polynomial p(x) of any degree with nonnegative integer coefficients. Bob can infer the coefficients of p(x) by only two evaluations as follows. He chooses a ...
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### Circular ring of 19 cells. Find minimum number of targets to make all inactive.

Imagine a circular ring of 19 cells. Each cell has a coronavirus inside it: either active or inactive. Scientists have developed a special drug that can target a specific cell (of your choice). When ...
49 views

### Guess if it's bigger or smaller number [duplicate]

Alice writes two distinct real numbers between 0 and 1 on two chits of paper and places them in two different envelopes. Bob selects one of the 2 envelopes randomly to inspect it. He then has to ...
29 views

### A regular hexagon is tiled with diamonds in three orientations. Prove that the same number of tiles appear in each orientation. [duplicate]

A large regular hexagon is cut out of a triangular grid and tiled with diamonds (pairs of triangles glued together along an edge). Diamonds come in three varieties, depending on orientation; prove ...
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### Ants on a 5-foot pole with a middle ant

Alice starts in the middle of a 5 foot pole. There are n other ants placed randomly on the pole and they start scampering in random directions with constant speed of 1 inch per second. Alice has a ...
1 vote
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### How to solve a prime number-logic problem

I am sitting with two other people who are wearing hats with the prime numbers 5 and 11. My hat has a prime number also, but I don't know what it is, nor do each of them know their numbers. It is also ...
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### 2 dice -> 9 cards

Alice has two standard dice with labels 1 thru 6. When she rolls them and adds their labels, she gets a distribution over integers in [2, 12]. Bob has nine cards, each labeled with some real number. ...
77 views

### Maximum tiling by Y Hexomino

"Y Hexomino" has a shape as shown in the picture. What is the maximum number of Y Hexomino that can be placed on a $13\times 13$ chessboard, where each Hexomino does not overlap? From the ...
44 views

Imagine a cube on a flat table, tantalizingly balanced on one of its vertices such that the vertex most distant from it is vertically above it. What will be the projection on the table if there is a ...
187 views

### Expected value with unit cubes and random lines

I am trying to solve this month's Jane Street puzzle in which there is a 3-space partitioned into unit cubes and a line of length D is randomly chosen uniformly in location and orientation. I want to ...
41 views

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### Lightbulb riddle

Suppose you have $n$ lightbulbs evenly distributed over a circle. Each lightbulb can have two states; it's either turned on or turned off. Next to each lightbulb there is a button that changes the ...
1 vote
119 views

### The toys problem: Probability of getting two matching good item and a different third Item

I've encountered an intriguing probability problem. I just registered to ask this, so this will be my first post. Disclaimer: I met this problem in a real setting that's it too convoluted to explain (...
45 views

### What are way to represent Zebra(Einstein) Puzzle mathematical or formal form

I look how to implement the algorithm (program) for solving so called Zebra(Einstein) Puzzle. For that, i look for some formal or mathematical representation of conditions in this puzzle. Can you ...
68 views

### find the shaded numbers that make a grid puzzle ready and complete the puzzle

A 5 by 5 Latin Square is a 5 by 5 grid of squares in which each square contains one of the numbers 1 through 5 so that every number appears exactly once in each row and column. A partially completed ...
412 views

### Placing the $21$ two-digit primes into a grid, such that primes in adjacent squares have either the same tens digit or ones digit

This is USAMTS round 3, problem 1 of the 2020-2021 Academic Year. Place the 21 2-digit prime numbers in the white squares of the grid on the right so that each two-digit prime is used exactly once. ...
204 views

### Minimum swaps to put an array into desired order, where some elements are identical/repeated

Inspired by a word game Waffle, see footnotes if interested. The abstracted problem: You're given an input array of letters, some of which might be identical (i.e. repeated), e.g. ...
49 views

### Jigsaw puzzle ratio question.

I was doing a puzzle when this question popped into my head: Assuming a puzzle had an equal amount of pieces in each row, and an equal amount of pieces in each column, what would be the least amount ...
94 views

### Expected time until you reach the top of staircase in a dice game

I received this question during an online assessment and it continues to bug me. The setup is the following: you have a staircase with $100$ stairs in front of you, and at each time step $t$ you roll ...
1 vote
105 views

### Fox and rabbit on circular path

Thank you for your solutions "Can fox catch rabbit" (There are five holes in a line. One of them is occupied by a rabbit. Each night, the rabbit moves to a neighboring hole, either to the ...
59 views

### Finding defected boxes puzzle

If I have say $6$ boxes of marbles such that each of the box contains $100$ marbles each of which weigh $1$ gram. Now I get a call from my marble dealer saying that there is a chance that any of the ...
1 vote