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Questions tagged [puzzle]

This tag is meant for questions about the mathematical principles behind games, riddles, or their possible solutions. If the answer is known to you please do not use this tag to "riddle" other users, but rather to ask about the correctness of a possible solution or ways to extend and improve an existing solution.

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343 views

Infinite Rubik's Cube

Is there an infinite analog to the Rubik's Cube? What does its solution-algorithm look like? For illustration, consider the Rubik's cube with infinite tiles to a side, on all sides, with sides of ...
22
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0answers
2k views

This should be a piece of cake… right?

You probably know the following problem: We have two circular cakes of the same height but unknown and potentially different radii, and we want to cut them into two equal shares. Each cut can only ...
13
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0answers
378 views

History of a combinatoric problem: exchanging numbers by throwing stones

Another user recently asked a question on the Puzzling stack: Two spies throwing stones into a river. Suitably generalised, it becomes: Two spies (Alice and Bob) need to exchange a message. Each ...
12
votes
0answers
402 views

How many distinct Unruly boards are there?

Unruly is a puzzle game played on a board of $2n × 2n$ squares. Each square must be colored either black or white, with the following constraints: Each column must have $n$ white squares and $n$ ...
9
votes
0answers
373 views

$5 \times 5\;$ “square additive set”

Problem: IBM Research - Ponder This - January 2019 monthly contest (which was closed few days ago) leads to the problem: Find sets $A = \{a_1,a_2,\ldots,a_n\}$, $B = \{b_1,b_2,\ldots, b_m\}$ such ...
9
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0answers
198 views

Solving general (dis)entanglement puzzles

What is the state of the art in (modelling and) solving a general (dis)entanglement puzzle? The following picture shows a nice example: There is a project called "The Untangler", which seems to be ...
8
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0answers
308 views

Separating Heavier from the Lighter Balls

This was posted Here and received a good answer, solving the general questions in either $n$ or $n+1$ moves, which is by just half a move on average "less good" than my manual solutions here. Classic ...
8
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0answers
231 views

Limit approximation for $\pi$ in the four fours puzzle?

The four fours puzzle is a recreational math puzzle whose aim is to express whole numbers using four occurrences of the digit 4 and a specified set of operators. A common variety permits the following:...
7
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0answers
97 views

Can this puzzle be solved using the representation theory of quivers?

This riddle originates in the youtube video here. It's mathematical content was summarised here as follows: There's a $5\times 5$ grid of nodes, all nodes are (bidirectionally) connected to their ...
7
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0answers
78 views

Flipping odd number array by two numbers in series.

Suppose an array of length $n$(odd number) which is composed with only $0$. We start to ‘flip’( or ‘change’) numbers with following rules. [1, 1] → [0, 0] [0, 0] → [1, 1] [1, 0] or [0, 1] : do not ...
7
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0answers
250 views

How to prove if an N size jigsaw puzzle is solvable.

Let's say we have a jigsaw puzzle with N pieces. Each piece has 4 sides and can only have one of three fits: a straight fit, a concave fit and convex fit. A side can only fit with another side if both ...
6
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0answers
142 views

Impossible permutations of the Gear Cube

If you're familiar with the group properties of the Rubik's Cube, you will probably know that, under the action of the standard moves, all possible permutations of the (unoriented) edge pieces are ...
6
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0answers
134 views

Coin weighing to find $k>3$ similar-weight sets

Fix $k>3.$ There are $n$ coins with positive real weights. You have a scale that, between two sets of coins, tells you which set is heavier, or if they are equal. Is it possible to perform at most $...
6
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0answers
275 views

Worst case in decanting puzzles (pouring water from one jug to others).

A classic puzzle is to start with $3$ jugs of nonzero integer capacity ($A \ge B \ge C$) and have some water (integer) in each jug (the initial position). The goal is to get to some final (integer) ...
6
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0answers
413 views

A product puzzle

This is from a math contest. I have solved it, but I'm posting it on here because I think that it would be a good challange problem for precalculus courses. Also, it's kind of fun. Write the ...
5
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0answers
54 views

Knights and knaves on a square grid

Today Gathering For Gardner posted a video by Yoshiyuki Kotani called "Liar/Truth Teller Patterns on Square Planes". The idea is that you fill a grid with knights and knaves so that both the knights ...
5
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0answers
78 views

A coding theory/probability puzzle

I thought of the following problem and I am stuck in solving it. Suppose there is a deck of 4 cards with 2 red and 2 blue. I pick 2 cards at random and choose 1 and show the other to my friend. With ...
5
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0answers
69 views

Flip $n$ coins on a circle. Assume a coin has been chosen from among those whose neighbors are both heads. What's the probability it is heads?

This is a generalization of the problem below (first appeared here) I am particularly curious to know if there is a closed-form formula to calculate the probability for any $n$ and any probability of ...
5
votes
0answers
101 views

Permutations of Rubik's cube such that no adjacent sticker is the same

I've always wondered, what is the number of possible permutations of the Rubik's cube such that any two adjacent stickers has a different color. By a permutation I mean a configuration of the cube ...
5
votes
0answers
330 views

Fixing a wobbly table - Revisited

There is this video on the Youtube channel Numberphile in which it is argued that each wobbly four legged square table can be made stable by just minor adjustments in position, namely an at most $90^°$...
5
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0answers
357 views

A puzzle with some jumping frogs

(The following puzzle is ispired by this nice video of Gordon Hamilton on Numberphile) In a pond there are $n$ leaves placed in a circle, for convenience they are numbered clockwise by $0,1,\ldots,n-...
5
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0answers
149 views

Subdividing regular pentagon into six congruent pieces

It is easy to divide a 2-gon into 3 congruent line segments. It is also easy to divide a triangle into 4 smaller triangles that are congruent. One of Martin Gardner's favorite problems is to show that ...
5
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0answers
235 views

Adding Numbers Pattern

A few nights ago I couldn't sleep and so started doing this: I would take a number and add up all of its digits to get a new number and then add up all of those digits and so on until there was only ...
5
votes
0answers
307 views

Puzzle - zero knowledge proof

I am solving the following problem : I have edge-matching puzzles, where all pieces are squares and the grid has $n$*$n$ format. There is no global image to guide a puzzle solver. Despite the puzzles ...
5
votes
0answers
172 views

Find the number that follows the rules of two different series

I have this logic problem where I need to find a number that fits in 2 different series (vertical and horizontal). Each series has a rule, and once you find them, you can determine the answer. $$ ...
4
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0answers
45 views

Minimum number of dominoes on an $n \times n$ chessboard to prevent placement of another domino.

OEIS sequence A280984 (based on this Math Stack Exchange question) describes the minimum number of dominoes on an $n \times n$ chessboard to prevent placement of another domino. The sequence ...
4
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0answers
64 views

$3$ scorpions are chasing $1$ ant on the edges of a cube. The ant is $3$ times faster than any scorpion. Can the ant survive?

The problem: Three scorpions are chasing a single ant on the edgegraph of a cube. The scorpions have the same speed ($v$), while the ant is $3$ times faster ($3v$). They can move in any direction and ...
4
votes
0answers
204 views

Graph Envelope Constraint puzzles from The Witness game.

The computer game "The Witness" contains various puzzles based on a finite square grid graph arranged in the usual way. A path must be found from a given point on the edge to another. Each square can ...
4
votes
0answers
972 views

Number of ways to color a grid?

I have a $N \times M $ grid and I am trying to calculate the number of ways I can color this grid in maximum $k$ colors (I can use only $2$ colors or all $k$ colors) with the exception that two ...
4
votes
0answers
160 views

A fashion victim puzzle

Consider $n \in \mathbb{N}$ fashion-sensitive kids, each wearing a T-shirt; for simplicity, kid $i \in \{1, \ldots, n\}$ initially wears shirt $i$. Tastes over the shirts are summarized in an $n \...
4
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0answers
103 views

What's the geometry of a puzzle key called?

Is there a name for the geometry of a classic puzzle key? It's not an ellipse, neither a circle, ...
4
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0answers
196 views

Is it a “paradox”, or a flaw in the question?

(Clearly not a pardox per-se but I would like to hear what you think) The basic riddle (not a very interesting one even) goes as follows: A first client comes into a barber shop, takes a hair cut ...
4
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0answers
51 views

Given $n$, find $a,b$ such that $a+b=n$ and $\Omega(a)+\Omega(b)$ is maximized

Given a number $n$, find $a,b$ such that: $a,b$ non-negative integers $a+b=n$ $\Omega(a)+\Omega(b)$ is maximized $\Omega(n)$ counts the number of prime factors of n (with multiplicity). For ...
4
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0answers
844 views

What is the highest possible score in 2048 hard?

There is a variant of the popular game 2048, called 2048 hard or 2048 impossible, which automatically places each new tile in the hardest possible location. Is this variation possible to solve, and if ...
4
votes
0answers
91 views

Would you please take a look if my substantiation is correct?

The four numbers 4, 5, 6, 7 are randomly inserted into 7 .3 .4 . 6 . 48 The result is a ten-digit number - for example, 7 4 3 5 4 6 6 7 48 How high is the chance, that the number created is ...
4
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0answers
675 views

Can this be only solved by trial and error?

The following question was asked in a competitive exam Rectangular tiles each of size 70 cm by 30 cm must be laid horizontally on a rectangular floor of size 110 cm by 130 cm, such that the ...
3
votes
0answers
57 views

Catching the mole problem best winning strategy

The problem is: There is a mole and n holes named with numbers $1, 2, 3, \ldots, n-1, n$. The mole can start from any hole and each day it can move only from a hole to a consecutive hole. So if the ...
3
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0answers
71 views

Puzzle (1): Explaining a pattern in multiplication graphs modulo $m$

In the plots of multiplication graphs $\mu^n_m(k) = kn\ \%\ m$ (with $a\ \%\ b$ meaning $a$ modulo $b$) the dots are undeniably arranged along parallel lines with a well-defined common slope $\alpha$ ...
3
votes
0answers
58 views

Maximum number of umbrellas that can be added in a one kilometer beach?

Suppose we have a beach of length $1-$km. Suppose one Day $0$, the beach is empty. One day $1$, a family comes and puts their umbrella at some point in the beach. This point is fixed forever and ...
3
votes
0answers
159 views

Explaining MENSA's IQ Problem on Alien Fingers

There is one probably quite well-known IQ puzzle below from Mensa: There are a number of aliens in a room. Each alien has more than one finger on each hand. All aliens have the same number of ...
3
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0answers
43 views

Q*Bert minimum moves to solve level

In the old video game Q*Bert, Q*Bert hops on squares on a 7-high pyramid to try to change them a certain color. Q*Bert can also jump off the edge onto discs, which drop him back on the top square, ...
3
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0answers
81 views

Number of divisors 888,888.

For an hour, the number of questions at math.stackexchange.com will reach 888,888. This number has exactly $2^7 = 128$ divisors. Which next integer will have the same number of divisors?
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0answers
96 views

Is it possible to write every integer with only one $3$?

There's a cool math challenge about writing every number from 0 to 100 with exactly three $3$'s. Most of the solutions use a clever way to write $50$ with only one $3$ : $$ \left \lfloor\sqrt{\sqrt{...
3
votes
0answers
165 views

Gauss-Jordan elimination on banded matrix $O(n)$ time?

My motivation is solving arbitrary Lights Out puzzles with a method given in MathWorld. Normally Gauss-Jordan elimination takes $O(n^3)$ time, but notice the addition matrix that combines all the "...
3
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0answers
151 views

Find four whole numbers… Is this maths question impossible?

I'm helping out my nephew with his 11+ exam revision. We have practice papers but no answers so. One of the questions is as follows: Find four whole numbers whose sum is 400, such that the first ...
3
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0answers
387 views

A list of math puzzle sites that require little knowledge beyond first year college math, just a lot of creativity?

A while back I found IBM's Ponder This website. http://www.research.ibm.com/haifa/ponderthis/ What is wonderful about that site is that a large number of the problems there don't require advanced ...
3
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0answers
213 views

Extending the general solution to the “Four fours” problem

The four fours puzzle is, given a number $n$, how can you represent $n$ with common (or more specifically, elementary) mathematical functions and 4 or less occurrences of the digit 4. On the ...
3
votes
0answers
200 views

Candy Crush as an integer programming problem

I'm trying to model the basic version of a match-three game, where the player (has a maximum number of swaps) must swap any two adjacent gems (no diagonals) in an 8x8 grid of gems in order to match ...
3
votes
0answers
256 views

When solving a big Rubik cube (100x100x100), do you reduce the solution to like 50x50x50, and then 25x25x25, and then like 10x10x10 and then 3x3x3?

My question is about Rubiks cube. Say you're solving a 100x100x100 cube (you can see examples in youtube by computer program - https://www.youtube.com/watch?v=0cedyW6JdsQ) When solving a big Rubik ...
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0answers
147 views

Math puzzle I have been stuck on

Have had this math puzzle that I have been unable to solve for a while. Each leter is a number between 1-9. No letter uses the same number twice (aka if B is 3 D can't be 3 also). The ? mark ...