# Tag Info

Accepted

• 76.3k
Accepted

### A colorful dodecahedron

Partial Answer: Solution: Other Solutions: Fun Stuff:
• 8,071
Accepted

### Do Langford squares exist?

Langford squares are not possible. Consider the middle two rows of a $2n \times 2n$ Langford square. They, like all rows, must contain $n$'s. Any $n$ must have a partner $n$ in its column that's $n+1$ ...
• 26.3k
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### Selectively neglected collection

These mannequins all have something in common... So what we need to do to assemble the remaining mannequins is too... And as tmpearce points out the comments,
• 6,683
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### The shorter the message, the larger the prize

Andrei can send a message that is: How?
• 14.2k
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### 5 chess pieces dominating a 5x5 grid

This arrangement of pieces should work:
• 5,632
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### Wizard of subsets

Unless I'm missing something
• 5,793

### Wizard of subsets

Move those marked l to the left, then the rs right and then the us up.
• 19.1k

### Wizard of subsets

Very similar to @loopy-walt solution, but still OC :)
• 769
Accepted

### What is the number of ways to spell French word « chrysanthème »?

The word can be spelled First we Then we If we
• 18.3k
Accepted

### Playing Mastermind against an angel and the devil

For the angel: For the devil:
• 14.2k

### The shorter the message, the larger the prize (version II)

I will give a very weak lower bound (improved based on xnor's idea), and a minor improvement to user1502040's answer. Lower bound: Proof: 11111111111111111111 ...
• 2,976

### 7 mathematicians around the clock in prison

HEADS UP: If you find this post interesting in principle but too terse or messy for your liking @Andrew Savinykh has written a more friendly version. UPDATE: Here is a program demoing the strategy ...
• 19.1k
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### 7 mathematicians around the clock in prison

This is the same solution by loopy-walt, presented a bit differently and adding some context. It seemed to me that the solution has some leaps that may be obvious to many people but also could be ...
Accepted

### Creating a clever hemisphere

This is a standard application of the pigeonhole principle.

### The shorter the message, the larger the prize (version II)

It is possible for them to make
• 677
Accepted

### Cable with mixed wires

Asymptotic result Lower bound Upper bound
• 19.1k
Accepted

### Clock hands get it Right

Here is a way I like to think about it Where does your reasoning fall down
• 133k
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### Do non-trivial Skolem squares exist?

No non-trivial Skolem squares exist. First, observe that any number $k$ appearing in a Skolem square must be part of an axis-aligned square of width and height $k$ whose four vertices are all $k$. For ...
• 26.3k

### 5 chess pieces dominating a 5x5 grid

This also works: Thanks @justhalf, this is works:
• 4,834

### 5 chess pieces dominating a 5x5 grid

Came up with an alternative solution from the other answers
• 301

### 16 queens puzzle

For the question of maximizing the number of sets of $8$ queens, define a graph with a node for each of the $92$ solutions for $8$ queens and an edge for each pair of solutions that share a cell. Now ...
• 12k
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### Save now! All the digits at half the price

Let's define some terms: Now we can begin. Let's break the problem down now. Now for the final count!
• 8,317

### Two arcs equal three arcs

Here's one solution. See this album for larger images. The multiset sum of the two arcs on the left, when placed on top of each other, equals the multiset sum of the three arcs on the right when ...

### Playing Mastermind against an angel and the devil

I don't know if it's optimal, but the devil can show at least I determined this by
• 25.7k
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### Good and bad numbers of remaining mines

Seems pretty straight forward: P.S.
• 2,963
Accepted

### n*n*n Rubik's cube algorithm

For a cube of edge n: n=1 is trivial. n=3 can be solved with known algorithms, such as the CFOP method. n=2 has faster algorithms but can also be solved as just the corners of the n=3 case. For n&...
• 76k
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• 3,035
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### Visiting all strings by swapping

By introducing a dummy node that is adjacent to all other nodes, we have a traveling salesman problem on a graph with $37$ nodes and $130$ edges. The distance between nodes is $0$ if either node is ...
• 12k
An easy lower bound is The minimum turns out to be You can solve the problem via integer linear programming as follows. Let $C=\{1,\dots,100\}$ be the set of coins. Let \$P=\{(i,j): 1 \le i \le j \...