# Tag Info

Accepted

### Professor Halfbrain and the sum of the digits of all divisors

Professor Halfbrain's theorem is Proof
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### Desegregate the Knights

Give these names to all the squares: 163 4 8 725 Each number can only be accessed by way of the numbers before and after it (where 8 wraps around to 1). That ...

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### The last number on the blackboard

Parity If you remove any two even numbers, their difference will also be an even number. The total number of odd numbers will remain the same. If you remove any two odd numbers, their difference ...
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### Averaging numbers on the blackboard

First choose $2014$ and $2016$. Average = $2015$. Now take the $2015$s. Their average is $2015$. Now choose $2015$ and $2013$. Average = $2014$. Choose $2014$ and $2012$. Average = $2013$. Note ...

### Which is larger? $\sqrt{15} - \sqrt{7} + \sqrt{5} + \sqrt{2}$ versus $5$

The answer is I imagine the line of reasoning the author wants is as follows:

### The last number on the blackboard

Building on @2012rcampion's answer, here's a simple constructive proof that any even number is reachable. Take the (even) number that you want to reach (say $n$) and put it aside. Then take the ...

### Is it possible that the last piece the ant has eaten is the central one?

I will give you a hint rather than an answer (because I think you will enjoy this more by solving it yourself): I would encourage others not to post an outright solution, at least for a few days; ...
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### Exterminating blobs on a grid

Given an arrangement of blobs, how can you determine whether it is possible to exterminate them all? What strategy can you use to succeed when possible? Warning: what follows is a constructive but ...

### Amnesiac in a ring shaped palace

Edit: A lot of credit is due to @ffao for devising a better way to deal with the case where there is just one room and reducing the solution by one. (Subsequently, @Lawrence has managed to do even ...
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### Amnesiac in a ring shaped palace

N = 2 By referencing the wall colors, you should be able to deduce which direction you were going (CW/CCW or R/L) when entering your current room. I used that assumption to come up with a state ...
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### Another curious incident in the flea circus

This answer is entirely due to Henning Makholm. If we draw a grid in such a way that the four fleas are at positions $(0,0), (0,1),(1,0)$ and $(1,1)$, then the fleas will forever be at integer points ...

### Will you be the first to get free?

There is a simple solution. Because
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### One hundred tiles

Label the tiles like chessboard notation, so that the bottom left is a1 and the top right is j10. Moves will be denoted by referring to a pair of opposite corners in the square being flipped. This ...
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### Consecutive Towers of Hanoi

Here is a revised solution, for... ...which  (again) seems like the maximum to me.  has been verified by Molhan as being maximal.   Trivial steps have been condensed. These ...

### Amnesiac in a ring shaped palace

N=5 This builds on @ffao's on-on pattern and is a slight optimisation over @hexomino's 7-step 6-step solution. I'm assuming that in a 1-room scenario, stepping out of that room will lead to the ...
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### Numbers on the blackboard: From 2-2015 to 1-2014

Answer The (maybe) interesting details The ugly details
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### Sliding balls on a 5x5 grid

This is definitely a bit more difficult that the ones before, but despite OP's warnings in the comment section, there's no need for any kind of brute force. (There's an answer with a solution already, ...
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### Put the colours back in order

Assuming that the starting position is the one shown in the screenshot, all of the colors are already in the bar, so the only possible move is to swap a color with the first one in the bar. To put a ...