# Tag Info

Accepted

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

Professor Halfbrain's theorem is Proof
• 136k
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 ...
• 1,348

### 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:
• 136k

### 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; ...
• 120k
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 ...
• 356
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 ...
• 15.4k

### 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 ...
• 136k

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

There is a simple solution. Because
• 30.6k

### 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 ...
• 7,929
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 ...
• 21.9k
Accepted

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, ...
• 77.6k

### Averaging numbers on the blackboard

Reasoning: After this we... ...going all the way back to...
• 709

### 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....
• 8,640

### 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 ...
• 77.6k
Accepted

### Blackboard problem with polynomial

The smallest possible value of $n$ is Claim: We can get every non-negative integer $n\leq 2016$ on the board. Proof: By induction. We start with $n=0$ on the board. We can get $1$ using Lord of the ...
• 14.3k
Accepted

### Sliding balls on a 4x4 grid version 2

Is this a trick question? (New here) Is the answer just Otherwise,
• 206

### Amnesiac in a ring shaped palace

[EDIT: hexomino points out in their answer that we can just use a completely off pattern instead of an alternating pattern to eliminate most of the states here, which makes this answer unnecessarily ...
• 21.8k
Accepted

Method:
• 9,917
Accepted

### Aatif averages numbers on the blackboard

The answer is : Explanation : Generalization :
• 7,165
Accepted

### Four indeed is cosmic!

Let the starting number be $n$. Consider the case where $n < 10$.  \begin{align} 1 &\to 2 \to 4 \\ 2 &\to 4 \\ 3 &\to 6 \to 12 \to 24 \to 2 \to 4\\ 4 \\ 5 &\to 10 \to 1 \to 2 \to ...
• 7,929
Accepted

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

A cube of dimension $3×3×3$ is made of sugar and consists of 27 small cubical sugar pieces arranged in the $3×3×3$ pattern. An ant is eating the sugar in such a way that it starts at one of the ...
• 3,219
Accepted

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

Yes. Explicit Grid
• 2,260

### Consecutive Towers of Hanoi

I wrote a program to find the answer to this question. I indeed found that 31 was the maximum number of disks for a 6 peg board. I also ran the program with 3, 4, and 5 pegs. Interestingly, the ...
• 61

### Professor Halfbrain and the 52 cards

The optimal solution is: First an example of the algorithm with 10 elements: The algorithm used follows: I will also only amend to the other solutions proof to show that the total number of swaps ...
• 61

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

Yet another method: And then:
• 494
Accepted

### Labyrinth of Teleporters

Without the pebble... But... Using the pebble...
• 3,316

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

This can be done with various approaches, I have tried using a very simple logical approach here without any previous knowledge required (only a pencil & paper)
• 372
Accepted

### Is it possible (for some configuration of initial 9 flowers) to get all red flowers after finitely many years?

The answer is: Slightly more rigorously, for all the flowers to be red: So at some point: Then: Then: But then: So: Note:
• 22k