Pythagorean Ant
Dave is doing his homework on the balcony and, preparing a presentation about Pythagorean triangles, has just cut out a triangle with side lengths 30cm, 40cm and...
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John Rhodes
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August 16, 2023
Friend Numbers
Let’s call two numbers friend numbers if their representation in base $10$ has at least one common digit. E.g. $1123$ and $3981$ are friend numbers. Let $f(n)$...
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John Rhodes
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August 16, 2023
Hallway of Square Steps
Peter moves in a hallway with $N + 1$ doors consecutively numbered from $0$ through $N$. All doors are initially closed. Peter starts in front of door...
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John Rhodes
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August 16, 2023
Roman Numerals II
A random generator produces a sequence of symbols drawn from the set {I, V, X, L, C, D, M, #}. Each item in the sequence is determined...
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John Rhodes
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August 16, 2023
$\pi$ Sequences
For every $n \ge 1$ the prime-counting function $\pi(n)$ is equal to the number of primes not exceeding $n$. E.g. $\pi(6)=3$ and $\pi(100)=25$. We say that a...
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John Rhodes
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August 16, 2023
Divisor Sums
Let $D(m,n)=\displaystyle\sum_{d\mid m}\sum_{k=1}^n\sigma_0(kd)$ where $d$ runs through all divisors of $m$ and $\sigma_0(n)$ is the number of divisors of $n$. You are given $D(3!,10^2)=3398$ and $D(4!,10^6)=268882292$. Find...
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John Rhodes
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August 16, 2023
Marsh Crossing
Frodo and Sam need to travel 100 leagues due East from point A to point B. On normal terrain, they can cover 10 leagues per day, and...
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John Rhodes
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August 16, 2023
Pairwise Coin-Tossing Game
Consider an $n$-player game played in consecutive pairs: Round $1$ takes place between players $1$ and $2$, round $2$ takes place between players $2$ and $3$, and...