Swapping Counters
A horizontal row comprising of $2n + 1$ squares has $n$ red counters placed at one end and $n$ blue counters at the other end, being separated...
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John Rhodes
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August 15, 2023
Factorials Divisible by a Huge Integer
Let $N(i)$ be the smallest integer $n$ such that $n!$ is divisible by $(i!)^{1234567890}$ Let $S(u)=\sum N(i)$ for $10 \le i \le u$. $S(1000)=614538266565663$. Find $S(1\,000\,000) \bmod...
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John Rhodes
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August 15, 2023
Bounded Sequences
Let $x_1, x_2, \dots, x_n$ be a sequence of length $n$ such that: $x_1 = 2$ for all $1 \lt i \le n$: $x_{i – 1} \lt...
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John Rhodes
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August 15, 2023
2011 Nines
Consider the real number $\sqrt 2 + \sqrt 3$. When we calculate the even powers of $\sqrt 2 + \sqrt 3$ we get: $(\sqrt 2 + \sqrt...
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John Rhodes
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August 15, 2023
Firecracker
A firecracker explodes at a height of $\pu{100 m}$ above level ground. It breaks into a large number of very small fragments, which move in every direction;...
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John Rhodes
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August 15, 2023
Numbers in Decimal Expansions
Let $p = p_1 p_2 p_3 \cdots$ be an infinite sequence of random digits, selected from $\{0,1,2,3,4,5,6,7,8,9\}$ with equal probability. It can be seen that $p$ corresponds...
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John Rhodes
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August 15, 2023
Digital Root Clocks
Sam and Max are asked to transform two digital clocks into two “digital root” clocks. A digital root clock is a digital clock that calculates digital roots...
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John Rhodes
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August 15, 2023
The Mouse on the Moon
The moon has been opened up, and land can be obtained for free, but there is a catch. You have to build a wall around the land...