Can't cut squares from here (in the thickest Mainer accent imaginable)

Consider a generic square. It can be subdivided into smaller sub-squares, not necessarily of equal size so that the smaller squares do not overlap. For example, you can slice the big square into four smaller squares, each a quarter of the area of the big square. Or you could slice it into seven squares, if you take one of those four squares and slice it into four yet smaller squares.

What whole numbers of squares can you not slice the big square into?

Let $S$ be the set of whole numbers $n \in \mathbb{N},$ where the original big square can be partitioned into $n$ not necessarily equally sized smaller squares. The problem setup provides a hint that shows that for any $n \in S,$ by taking any single smaller square and dividing it equally into $4$ even smaller squares, we end up with $n + 3$ subsquares, so $n \in S \Rightarrow n+3 \in S.$ So in particular, since $1 \in S$ (just the original square with no further cuts), we have $(3\mathbb{N} + 1) \subset S.$

However, we needn't focus solely on subdividing into $4$ equally sized sub-squares. For instance for any $k \in \mathbb{N}$ we can subdivide into $k^2$ equally sized sub-squares by equally dividing the length and width of the original square into $k$ segments and forming a $k$ by $k$ grid of subsquares. In this case, for any $n \in S,$ we can take any sub-square and replace it with $k^2$ smaller sub-squares, getting $n + k^2 - 1$ total sub=squares. In particular, for $k = 3,$ we get $(8 \mathbb{N} + 1) \subset S,$ which implies that for instance $9 \in S$ and $17 \in S.$

Since $9 \equiv 0 \mod 3,$ by further subdividing squares into equal quarter squares, we get that any multiple of $3$ greater than or equal to $9$ is included in $S.$ That is, $(3 \mathbb{N} \setminus \{0,3,6\} ) \subset S.$ Similarly, since $17 \equiv 2 \mod 3,$ we get that $((3 \mathbb{N} + 2) \setminus \{2, 5, 8, 11, 14\}) \subset S.$ So since all remainders with respect to $3$ are covered for large enough values, we see that \begin{align*}S &= (3 \mathbb{N} + 1) \cup (3 \mathbb{N} \setminus \{0, 3, 6\}) \cup ((3 \mathbb{N} + 2) \setminus \{2, 5, 8, 11, 14\})\\ &= \mathbb{N} \setminus \{2, 3, 5, 6, 8, 11, 14\}.\end{align*}

Thus the complement of $S$ is given by $S^C = \{2, 3, 5, 6, 8, 11, 14\}.$

Attrition by subtraction

The Game of Attrition works by players successively subtracting their own remaining number of points from their opponents points until one player is left with a nonpositive point total. Assume player A has $N$ points to begin the game and always goes first against his opponent B.

What is the largest point total that B can start with such that A will still win?

Let's say that B has initial points of $P > N,$ since obviously if $P \leq N,$ then A will win in the first move. After one round of both A and B's moves, A will have $2N-P$ points remaining and B will have $P-N,$ both of which are binomials in terms of $N$ and $P.$

Let $A_n$ and $B_n$ be the point total for A and B, respectively after $n$ full turns. One full turn of gameplay gives the following recurrence:

\begin{align*} A_{n+1} &= 2 A_n - B_n \\ B_{n+1} &= B_n - A_n \\ A_0 = N, \,\,&\,\, B_0 = P\end{align*}

If we assume, as after the first turn, that both $A_n$ and $B_n$ are binomials in terms of $N$ and $P$, say $A_n = a_n N + b_n P$ and $B_n = c_n N + d_n P,$ then the above recurrence becomes \begin{align*} a_{n+1} &= 2 a_n - c_n \\ b_{n+1} &= 2 b_n - d_n \\ c_{n+1} &= c_n - a_n \\ d_{n+1} &= d_n - b_n \\ a_0 = d_0 = 1, \,\,&\,\, b_0 = c_0 = 0 \end{align*} We note that player $A$ will win after $n$ steps if $B_n \leq 0,$ or equivalently, if $$P \leq -\frac{c_n}{d_n} N.$$ The largest possible point total that B can start with such that A will still win is thus $$\sup \left\{ -\frac{c_n}{d_n}N : n \in \mathbb{N} \right\}.$$

Letting $x_n = (a_n, b_n, c_n, d_n)^T,$ and $$M =\left(\begin{array}{cccc} 2 & 0 & -1 & 0 \\ 0 & 2 & 0 & -1 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \end{array}\right),$$ we can express the reccurence more compactly in matrix notation as: $$x_n = Mx_{n-1} = \cdots = M^n x_0,$$ where $x_0 = (1, 0, 0, 1)^T.$ The matrix $M$ has eigenvalues $\lambda_\pm = \frac{3 \pm \sqrt{5}}{2},$ each of order $2,$ associated with eigenvectors $$u_\pm = \sqrt{\frac{2}{5\mp \sqrt{5}}}\left( \begin{array}{c} 1 \\ 0 \\ \frac{1 \mp \sqrt{5}}{2} \\ 0\end{array} \right) \,\,\, \text{and} \,\,\, v_\pm = \sqrt{\frac{2}{5\mp \sqrt{5}}} \left( \begin{array}{c} 0 \\ 1 \\ 0 \\ \frac{1 \mp \sqrt{5}}{2}\end{array} \right).$$ So let's define $$L = \left(\begin{array}{cccc} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ -\frac{\sqrt{5}-1}{2} & 0 & \frac{\sqrt{5} + 1}{2} & 0 \\ 0 & -\frac{\sqrt{5}-1}{2} & 0 & \frac{\sqrt{5} + 1}{2}\end{array}\right),$$ $$D = \left(\begin{array}{cccc} \frac{5-\sqrt{5}}{2} & 0 & 0 & 0 \\ 0 & \frac{5 - \sqrt{5}}{2} & 0 & 0 \\ 0 & 0 & \frac{5 + \sqrt{5}}{2} & 0 \\ 0 & 0 & 0 & \frac{5 + \sqrt{5}}{2}\end{array}\right) = L^TL$$ and $$\Lambda = \left(\begin{array}{cccc} \frac{3+\sqrt{5}}{2} & 0 & 0 & 0 \\ 0 & \frac{3+\sqrt{5}}{2} & 0 & 0 \\ 0 & 0 & \frac{3-\sqrt{5}}{2} & 0 \\ 0 & 0 & 0 & \frac{3-\sqrt{5}}{2}\end{array}\right).$$ Then the eigendecomposition of $M$ can be given by $M = Q \Lambda Q^T,$ where $Q = LD^{-1/2}.$ In particular, for any integer $n \geq 1,$ $$M^n = Q\Lambda^n Q^T = (LD^{-1/2})\Lambda^n (LD^{-1/2})^T = L D^{-1} \Lambda^n L^T.$$

For any integer $n \geq 1,$ $$D^{-1} \Lambda^n L^T x_0 = \left( \begin{array}{c} \frac{2}{5-\sqrt{5}} \left( \frac{3+\sqrt{5}}{2} \right)^n \\ -\frac{\sqrt{5}}{5} \left( \frac{3+\sqrt{5}}{2} \right)^n \\ \frac{2}{5+\sqrt{5}} \left( \frac{3-\sqrt{5}}{2} \right)^n \\ \frac{\sqrt{5}}{5}\left( \frac{3-\sqrt{5}}{2} \right)^n\end{array} \right).$$ So the recurrence relationship can be written as $x_n = M^n x_0 = L D^{-1} \Lambda^n L^T x_0,$ which is given by \begin{align*}x_n = M^n x_0 &= \left( \begin{array}{cccc} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ -\frac{\sqrt{5}-1}{2} & 0 & \frac{\sqrt{5} + 1}{2} & 0 \\ 0 & -\frac{\sqrt{5}-1}{2} & 0 & \frac{\sqrt{5} + 1}{2}\end{array} \right) \left( \begin{array}{c} \frac{2}{5-\sqrt{5}} \left( \frac{3+\sqrt{5}}{2} \right)^n \\ -\frac{\sqrt{5}}{5} \left( \frac{3+\sqrt{5}}{2} \right)^n \\ \frac{2}{5+\sqrt{5}} \left( \frac{3-\sqrt{5}}{2} \right)^n \\ \frac{\sqrt{5}}{5}\left( \frac{3-\sqrt{5}}{2} \right)^n\end{array} \right)\\ &= \left( \begin{array}{c} \frac{2}{5-\sqrt{5}}\left(\frac{3+\sqrt{5}}{2} \right)^n + \frac{2}{5+\sqrt{5}}\left(\frac{3-\sqrt{5}}{2} \right)^n \\ -\frac{\sqrt{5}}{5} \left(\frac{3+\sqrt{5}}{2} \right)^n + \frac{\sqrt{5}}{5} \left(\frac{3-\sqrt{5}}{2} \right)^n \\ -\frac{\sqrt{5}}{5} \left(\frac{3+\sqrt{5}}{2} \right)^n + \frac{\sqrt{5}}{5} \left(\frac{3-\sqrt{5}}{2} \right)^n \\ \frac{\sqrt{5}(\sqrt{5}-1)}{10} \left(\frac{3 + \sqrt{5}}{2}\right)^n + \frac{\sqrt{5}(\sqrt{5}+1)}{10} \left(\frac{3 - \sqrt{5}}{2}\right)^n \end{array} \right).\end{align*} Therefore, we have $$-\frac{c_n}{d_n} N = \frac{ \frac{\sqrt{5}}{5} \left(\frac{3+\sqrt{5}}{2} \right)^n - \frac{\sqrt{5}}{5} \left(\frac{3-\sqrt{5}}{2} \right)^n }{ \frac{\sqrt{5}(\sqrt{5} - 1)}{10} \left(\frac{3+\sqrt{5}}{2} \right)^n + \frac{\sqrt{5}(\sqrt{5} + 1)}{10} \left(\frac{3-\sqrt{5}}{2} \right)^n } N,$$ which is a monotonically increasing sequence with $$\lim_{n\to \infty} -\frac{c_n}{d_n} N = \sup \left\{ -\frac{c_n}{d_n} N : n \in \mathbb{N}\right\} = \frac{\sqrt{5}+1}{2}N.$$ That is, if $P \leq \frac{\sqrt{5}+1}{2} N,$ then A will still win, or more specifically, the function $$P(N) = \left\lfloor \frac{\sqrt{5}+1}{2} N \right\rfloor$$ gives the maximal value that B can start with but lose to $A$, who starts with N and goes first.

Putting Santul through his paces

Santul completes two 20 mile training runs with different pace profiles:

1) A constant 9 minute mile pace (which is impressive both physically and for its consistency); and

2) His pace starts at 10 minutes per mile and then linearly decreases to 8 minutes per mile over the course of the race. So for instance, when Santul is halfway done with the race (in time, not distance), his pace is 9 minutes per mile.

The question is: which run did he complete faster and what were the two times?

Santul's 9 minute mile run will complete in $9 \,\text{min} \, / \, \text{mile}\, \cdot 20 \,\text{miles} = 180 \,\text{minutes}.$

Let's now look at the second run. Let $$p(t) = a + bt$$ be the linear function of Santul's pace at time $t$. From the initial data we know that $$p(0) = a = 10$$ and that if $T$ is that time that Santul finishes his second run, then $$p(T) = 10 + b T = 8.$$ So $a = 10$ and $b = -\frac{2}{T}.$

We can convert Santul's pace into distance by integrating as follows $d(s) = \int_0^t \,\frac{ds}{p(s)},$ so given the 20 mile course was finished in $T$, we have $$20 = \int_0^T \,\frac{ds}{10 - \frac{2t}{T}} = -\frac{T}{2} \ln (\frac{8}{10}).$$ Solving for $T$ we get $$T = \frac{40}{\ln 1.25} = 179.256804709...$$ which is just a shade under the constant pace time.

Von Neumann's fair coin simulator

For any $p \in (0,1)$, von Neumann showed that an unfair coin which shows heads with probability $p$ can be used to simulate a fair coin by flipping twice, ignoring/redrawing for HH or TT and associating HT with T and TH with H. The issue is that depending on the value of $p,$ one might need to ignore/redraw for HH and TT outcomes for a very large number of tries before getting either TH or HT and exiting.

For what values of $p \in (0,1)$ is it possible to simulate a fair coin in at most $N$ flips?

While the phrasing on the question is a bit vague, I will assume two main frameworks for what "possible" means:

1) For what values of $p \in (0,1)$ is the expected number of flips needed to exit less than or equal to $N$?; or

2) Given some value of $q \in (0,1),$ for what values of $p \in (0,1)$ does the probability that the simulation exists in less than or equal to $N$ exceed $q$?

Let $T$ be the number of flips of the van Neumann simulator needed to exit. The probability that the van Neumann simulator exits after in one flip is $\mathbb{P}\{T = 1\} = 2p(1-p).$ The probability for $k > 1$ that it exits in $k$ flips is $\mathbb{P}\{ T = k \} = 2p(1-p) \cdot (1 - 2p(1-p))^{k-1}.$ So the average is \begin{align*}\mathbb{E}[T] &= \sum_{k=1}^\infty k \mathbb{P} \{ T = k \} = \sum_{k=1}^\infty k 2p(1-p) (1 - 2p(1-p))^{k-1}\\ &= 2p(1-p) \frac{1}{(1 - (1-2p(1-p)))^2} = \frac{1}{2p(1-p)}.\end{align*}

So under framework 1, the question becomes solving for $p \in (0,1)$ such that $$\mathbb{E}_p [T] = \frac{1}{2p(1-p)} \leq N.$$ Thus the solution is $$p \in \left[\frac{1 - \sqrt{1-2/N}}{2}, \frac{1 + \sqrt{1 - 2/N}}{2} \right].$$

Under framework 2, the question becomes for some $q \in (0,1),$ solving for $p \in (0,1)$ such that $$\mathbb{P}_p \{ T \leq N \} = \sum_{k=1}^N 2p(1-p) (1 - 2p(1-p))^{k-1} = 1 - (1 - 2p(1-p))^N \geq q.$$ Since $2p(1-p) \in [0,0.5]$ for all $p \in (0,1),$ if $q > 1-2^{-N},$ then $\mathbb{P}_p \{T \leq N \} \lt q,$ for all $p \in (0,1).$ But for any $q \in [0, 1-2^{-N}],$ then $\mathbb{P}_p \{T \leq N \} \geq q$ for $$p \in \left[ \frac{1 - \sqrt{2 \sqrt[N]{1-q} - 1}}{2}, \frac{1 + \sqrt{ 2 \sqrt[N]{1-q} - 1}}{2} \right].$$

A Game of Hot Pumpkin, or How I Was Left Holding the Chinese Remainder

In a game of Hot Pumpkin with $61$ people, the players count off clockwise upward from $1$ to $N$, eliminating the $N$th player and then then beginning again with the player to the left of the most recently eliminated. The integer value of $N$ is mutually agreed upon prior to beginning the game.

I start and say $1$, with the first eliminated player sitting $18$ spots to my left. Then Ricky starts up the next round and the person $31$ spots to her left is eliminated next. Zach both begins and is eliminated in the third round.

Based on the first three eliminations, what is the smallest value of $N$?

This game sets up as an example of the Chinese remainder theorem, which states that the system of congruences involved in this game, namely \begin{align*} N & \equiv 19 \mod 61 \\ N & \equiv 32 \mod 60 \\ N & \equiv 1 \mod 59,\end{align*} has a single unique solution in $\mathbb{Z}_{59 \cdot 60 \cdot 61} = \mathbb{Z}_{215940}.$

For any two congruences, $N \equiv a_1 \mod n_1$ and $N \equiv a_2 \mod n_2,$ with $\text{gcd}(n_1,n_2) = 1,$ we can use Euclidean algorithm to calculate the integer solution to $$m_1 n_1 + m_2 n_1 = 1$$ that is asserted by B\'{e}zout's identity. From here, we note that $$N = a_1 m_2 n_2 + a_2 m_1 n_1 \mod n_1 \cdot n_2 $$ is a solution to both congruences.

So in particular, we see that $(m_1, m_2) = (1, -1)$ is a solution to $61m_1 + 60m_2 = 1$ and hence $$N = 19 \cdot (-1) \cdot 60 + 32 \cdot 1 \cdot 61 \mod 61 \cdot 60 = 812 \mod 3660$$ is a solution to the first two congruences of our game of Hot Pumpkin.

Continuing onward, we can use the extended Euclidean algorithm to find that the solution to $3660m_1 + 59m_2 = 1$ is $(m_1, m_2) = (-29, 1799).$ This gives the smallest possible solution to all three congruences as \begin{align*}N &= 812 \cdot 1799 \cdot 59 + 1 \cdot (-29) \cdot 3660 \mod 3660 \cdot 59\\ &= 86080352 \mod 215940 \equiv 136232.\end{align*}

Climbing the mountains of Tour de FiveThirtyEight

You and three other, equally talented bicyclists are competing to get up the mountain to accrue points: 5 for first, 3 for second, 2 for third and 1 for last places, respectively.
Since you are all equally talented, the relative rankings of your average speeds for this stage should all be equally likely outcomes. However, two of your competitors are on a team and through a clever use of drafting off of one another will be able to both finish at roughly the same time as the faster of the two teammates.

Given that two of your competitors are working together, what is your expected number of points on this stage?
Let's call your average speed $A,$ the average speeds of the two teammates $B$ and $B^\prime,$ and the average speed of other individual competitor $C.$ Given that there only $24$ possible, equally likely orderings of $A,$ $B,$ $B^\prime,$ and $C,$ we will go with brute force to calculate the expected number of points for this stage.

If $A > \max\{ B, B^\prime \}$, then all of the teammate's schemings are for nought, and you will get the place that you rightfully deserve. There are 6 orderrings with $A > C$ and $A > \max \{B, B^\prime\}$, where you are in first and get $5$ points.
There are another 2 orderrings where $C > A > \max\{B, B^\prime\}$, where you place 2nd and receive $3$ points.

Similarly, if $\min \{B, B^\prime\} > A,$ then the cooperative advantage is also nill from your perspective. These cases cover the 6 orderings where $C > A$ and $\min\{ B, B^\prime\} > A$ where you were in last place and receive $1$ point.

There are also 2 orderings where $\min\{B, B^\prime\} > A > C$ and you finish in 3rd place receiving $2$ points.

The teamwork only affects you when either $B > A > B^\prime$ or $B^\prime > A > B.$ There are four possible orderings with $A$ between $B$ and $B^\prime$ and $A$ the second largest number in $\{A, B, B^\prime, C\}.$ In this case, though rightfully you should have finished in second place the slower of $B$ and $B^\prime$ gets pulled ahead of you by the faster and you place in 3rd, receiving $2$ points.
Similarly, there are 4 possible orderings with $A$ between $B$ and $B^\prime$ and $A$ the third largest number in $\{A, B, B^\prime, C\}.$ In this case, though rightfully you should have finished in third place, the slower of $B$ and $B^\prime$ gets pulled ahead of you by the faster and you place last, receiving $1$ point.

Summing up all of the points, we get $$6 \cdot 5 + 2 \cdot 3 + 6 \cdot 1 + 2 \cdot 2 + 4 \cdot 2 + 4 \cdot 1 = 58 \text{points,}$$ for an average of $2.41\bar{6}$ points.

Meanwhile in the absences of the teammate shenaniganry, your expected points is the simple average of the points $\frac{5+3+2+1}{4} = 2.75,$ so teammates cost you only a third of a point.

Pickup basketball teams

The number of ballers from each of Blacksburg, Greensboro and Silver Spring who show up to a pickup basketball game each week is identically and independently distributed as the uniform distribution on $\{1, 2, 3, 4, \dots, N\},$ for some integer $N.$

What is the probability that, on any given week, it’s possible to form two equal teams with everyone playing, where two towns are pitted against the third?

Let $B,$ $G$ and $S$ be the number of players who showed up from Blacksburg, Greensboro and Silver Spring, respectively. Without loss of generality, let's assume that $S$ is the largest and we seek the number of configurations where $B+G=S.$

First note that if $S$ is the maximum, then no such configurations can occur if $S=1(=B=G).$ For any $S = 2, \dots, N,$ there are $S-1$ different combinations of integers $B$ and $G,$ such that $B+G=S;$ those being $(B,G) \in \{ (1,S-2), (2, S-3), \dots, (S-3, 2), (S-1, 1) \}.$ So the total number of games where $B+G=S$ is $$\sum_{S=2}^N (S-1) = \frac{N(N-1)}{2}.$$
From symmetry, we get three times this number (since we arbitrarily assumed that $S$ was the maximum, but it could equally likely be $B$ or $G$), so the total number of configurations where two equal teams where two towns are pitted against the third are $3 \frac{N(N-1)}{2}.$ The total number of possible configurations of players to show up on a given week is $N^3,$ so the probability is \[p_N = \frac{3(N-1)}{2N^2}.\]
In particular, if $N = 5,$ then the probability is $p_5 = 24\%.$