31 votes
Accepted

Professor Halfbrain and the sum of the digits of all divisors

Professor Halfbrain's theorem is Proof
hexomino's user avatar
  • 133k
30 votes
Accepted

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 ...
Deusovi's user avatar
  • 145k
28 votes

One hundred tiles

Answer: Argument: ============ ============ ============
Gamow's user avatar
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27 votes
Accepted

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 ...
2012rcampion's user avatar
  • 18.6k
27 votes
Accepted

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 ...
iamwhoiam's user avatar
  • 1,348
26 votes

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:
hexomino's user avatar
  • 133k
24 votes

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 ...
Dr Xorile's user avatar
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22 votes

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; ...
Gareth McCaughan's user avatar
19 votes
Accepted

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 ...
noedne's user avatar
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18 votes

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 ...
hexomino's user avatar
  • 133k
18 votes
Accepted

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 ...
ajee's user avatar
  • 346
17 votes
Accepted

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 ...
16 votes

Will you be the first to get free?

There is a simple solution. Because
Florian F's user avatar
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15 votes
Accepted

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 ...
f'''s user avatar
  • 33.6k
15 votes
Accepted

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 ...
humn's user avatar
  • 21.8k
15 votes

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 ...
Lawrence's user avatar
  • 7,909
15 votes
Accepted

Swapping registers in an old calculator

Task 1 was answered. Task 2:
Anders Kaseorg's user avatar
14 votes

Numbers on the blackboard: From 2-2015 to 1-2014

Answer The (maybe) interesting details The ugly details
Sleafar's user avatar
  • 18k
13 votes
Accepted

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, ...
Bass's user avatar
  • 76k
12 votes
Accepted

Professor Halfbrain and the right-angled triangles

Professor Halfbrain For integer $a$, $b\geq 0$, let $T_{a,b}$ denote the right triangle with side lengths $$ \frac{3^{a+1} 4^b}{5^{a+b-1}},\;\;\;\frac{3^a 4^{b+1}}{5^{a+b-1}}\;\;\;\frac{3^a 4^b}{5^{a+...
Julian Rosen's user avatar
  • 14.2k
12 votes
Accepted

Create an impossible knight transformation

All 64 knights are needed, in which case any setup is stuck. We prove that with 63 knights, any position is reachable from any other position. With 63 knights, the puzzle is much like a sliding ...
xnor's user avatar
  • 26.3k
12 votes
Accepted

Blackboard problem with 2016

Obviously we're never going to get an irrational number. So any number we get can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime integers ($q$ can be $1$, if the number is an ...
f'''s user avatar
  • 33.6k
12 votes

Averaging numbers on the blackboard

Reasoning: After this we... ...going all the way back to...
MichaelK's user avatar
  • 709
11 votes
Accepted

Blackboard problem with polynomial

By the rational root theorem, any integer root of the polynomial $$a_nx^n + \dots + a_1x + a_0$$ is a divisor of its constant term $a_0$, if $a_0 \neq 0$. We will never let $a_0 = 0$, since that ...
Lynn's user avatar
  • 2,485
11 votes

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

Here's another approach, which starts from smaller integer residuals than hexomino's solution. and Finally, bringing these lines of argument togegther, I admit it isn't a single chain of inferences....
Rosie F's user avatar
  • 8,299
10 votes

Concentrating tokens on an infinite board

General Proof Let $T_n$ be the number of tokens at a Chebyshev distance $n$ from X; fancy way of imagining squares around X of increasing distance $n$ from X. The values of $T_n$ are; $$T_1=8$$ $$...
Trenin's user avatar
  • 8,954
10 votes
Accepted

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 ...
f'''s user avatar
  • 33.6k
10 votes

Professor Halfbrain and the 52 cards

This is not a solution, but it is a demonstration that the answer is not 51.   The answer must be at least
f'''s user avatar
  • 33.6k
10 votes

Sliding balls on a 4x4 grid version 2

@user already posted a solution, mine differs slightly at the end: (spaces inserted at the points where I was mentally switching to another task; the final 5 moves are where the solution differs from ...
Bass's user avatar
  • 76k

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