Kaprekar Constant
$6174$ is a remarkable number; if we sort its digits in increasing order and subtract that number from the number you get when you sort the digits...
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John Rhodes
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August 15, 2023
One-child Numbers
We say that a $d$-digit positive number (no leading zeros) is a one-child number if exactly one of its sub-strings is divisible by $d$. For example, $5671$...
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John Rhodes
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August 15, 2023
Gnomon Numbering
For integers $m, n$ ($0 \leq n \lt m$), let $L(m, n)$ be an $m \times m$ grid with the top-right $n \times n$ grid removed. For...
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John Rhodes
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August 15, 2023
Uphill Paths
Let $n$ be a positive integer. Suppose there are stations at the coordinates $(x, y) = (2^i \bmod n, 3^i \bmod n)$ for $0 \leq i \leq...
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John Rhodes
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August 15, 2023
Circle and Tangent Line
Let $C$ be the circle with radius $r$, $x^2 + y^2 = r^2$. We choose two points $P(a, b)$ and $Q(-a, c)$ so that the line passing...
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John Rhodes
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August 15, 2023
Nim Extreme
Let $n$ be a positive integer. Consider nim positions where:There are $n$ non-empty piles. Each pile has size less than $2^n$. No two piles have the same...
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John Rhodes
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August 15, 2023
Admissible Paths Through a Grid
Let’s call a lattice point $(x, y)$ inadmissible if $x, y$ and $x+y$ are all positive perfect squares. For example, $(9, 16)$ is inadmissible, while $(0, 4)$,...
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John Rhodes
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August 15, 2023
Idempotents
If we calculate $a^2 \bmod 6$ for $0 \leq a \leq 5$ we get: $0,1,4,3,4,1$. The largest value of $a$ such that $a^2 \equiv a \bmod 6$...