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17 votes
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What is the maximum number of fish possible in your tank?

The maximum number is Reasoning:
Benjamin Wang's user avatar
8 votes
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Filling a rectangular grid with holes using tetrominoes

No, it isn't always possible. Consider this 5-by-3 grid with 3 cells painted black. The 12 white cells can't be split into 3 tetrominos because the four corner cells must be in different tetrominos --...
xnor's user avatar
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6 votes

Largest sequence of adjacent numbers less than 11 such that adjacent number divides the other

The "divisor" graph $G$ has node set $\{1,2,\dots,10\}$ and an edge $(i,j)$ if $i$ divides $j$ or $j$ divides $i$. Paths of length $9$ are easy to find. To show that $10$ is impossible, we ...
RobPratt's user avatar
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6 votes

Largest sequence of adjacent numbers less than 11 such that adjacent number divides the other

We work backwards from our constraints, starting from the most restrictive numbers. 7 can only go next to 1: 7 1 Next 5 and 9 only have 2 options, either 1 or 10 and 3, respectively. So one of these ...
fencer22's user avatar
6 votes
Accepted

Unpaired socks in my lap

Let $N$ be the number of distinct pairs of socks, let random variable $X_k$ be the number of socks drawn to get the first pair when $k$ socks are in your lap, and let $E_k=\mathbb{E}(X_k)$. We want ...
RobPratt's user avatar
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6 votes

3³+4³+5³=6³ Puzzle

An alternative 6-piece answer to the second question, which has the largest possible first and second piece:
Bubbler's user avatar
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5 votes

The Bogotá Marathon

Some comments on part (b) (Warning: highbrow mathematics ahead, done by someone not actually particularly expert in the relevant field. Non-mathematicians will be intimidated, actual experts will ...
Gareth McCaughan's user avatar
4 votes

Unpaired socks in my lap

We'll work a slightly more general case of having $N$ distinct pairs of socks total. The specific problem posed is $N=10$. Denote the state $n$ as the state of having $n$ socks in your lap. When you ...
DanDan面's user avatar
  • 1,044
4 votes

A 3 digit perfect square and its reverse are both perfect squares. What is the number?

kagami's user avatar
  • 246
4 votes
Accepted

Flip-Flop Firefly Frenzy

The merchant appears to have had the idea However, So, the merchant entered this pattern, using 30 drops:
Deusovi's user avatar
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3 votes

Prisoners and warden game, again?

As a brief example:
kagami's user avatar
  • 246
3 votes

A 3 digit perfect square and its reverse are both perfect squares. What is the number?

Let $[a_r,\dotsc,a_0]$ denote $\sum_{i=0}^r a_i10^i$ with $0\leq a_i\leq 9$. Let $0\leq n \leq 999$ be a perfect square, say $n=m^2$, where $m=[a,b]$. Note that $m\leq 31$, so that $0\leq a\leq 3$. ...
rts's user avatar
  • 31
3 votes

Unpaired socks in my lap

TL;DR Let's first consider the case with only 2 pairs. Now let's extend to three pairs. Let's do one more: 4 pairs. Now, let's generalize that into p pairs.
dvx2718's user avatar
  • 236
2 votes

Unpaired socks in my lap

Answered the bonus question since I started working on it before OP edited. Also, by misreading the question I did the work for 10 total socks, instead of 10 pairs of socks. Maybe someone can use this ...
gannolloy's user avatar
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2 votes

What is the maximum number of fish possible in your tank?

The maximum number is because the problem you enunciated is equivalent to
dr_'s user avatar
  • 1,875
2 votes

Unpaired socks in my lap

This answer solves the bonus question, of what the average maximum number of socks in your lap is. The answer is Python code here. I solved this with a dynamic programming method. Consider a complete ...
causative's user avatar
  • 121
1 vote

Largest sequence of adjacent numbers less than 11 such that adjacent number divides the other

I can get nine of the ten numbers: Explanation: I'm not aware of the exact mental process of how I found a solution, but this is the entire jotting of how it emerged: My proof of the maximum:
Weather Vane's user avatar
  • 14.5k
1 vote

The Bogotá Marathon

Part 1 solved. Part 2 partial. Edit
Weather Vane's user avatar
  • 14.5k

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