Prime Frog
Susan has a prime frog. Her frog is jumping around over $500$ squares numbered $1$ to $500$. He can only jump one square to the left or...
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John Rhodes
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August 15, 2023
Lowest-cost Search
We are trying to find a hidden number selected from the set of integers $\{1, 2, \dots, n\}$ by asking questions. Each number (question) we ask, has...
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John Rhodes
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August 15, 2023
Rooms of Doom
A series of three rooms are connected to each other by automatic doors. Each door is operated by a security card. Once you enter a room the...
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John Rhodes
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August 15, 2023
Modulo Summations
Let $a_n$ be a sequence recursively defined by:$\quad a_1=1,\quad\displaystyle a_n=\biggl(\sum_{k=1}^{n-1}k\cdot a_k\biggr)\bmod n$. So the first $10$ elements of $a_n$ are: $1,1,0,3,0,3,5,4,1,9$. Let $f(N, M)$ represent the number...
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John Rhodes
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August 15, 2023
Stone Game II
A game is played with two piles of stones and two players. On each player’s turn, the player may remove a number of stones from the larger...
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John Rhodes
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August 15, 2023
Building a Tower
Let $f(n)$ represent the number of ways one can fill a $3 \times 3 \times n$ tower with blocks of $2 \times 1 \times 1$.You’re allowed to...
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John Rhodes
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August 15, 2023
Bitwise-OR Operations on Random Integers
Let $y_0, y_1, y_2, \dots$ be a sequence of random unsigned $32$-bit integers (i.e. $0 \le y_i \lt 2^{32}$, every value equally likely). For the sequence $x_i$...
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John Rhodes
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August 15, 2023
Binomial Coefficients Divisible by 10
Let $T(m, n)$ be the number of the binomial coefficients $^iC_n$ that are divisible by $10$ for $n \le i \lt m$ ($i$, $m$ and $n$ are...