Range Flips
$N$ disks are placed in a row, indexed $1$ to $N$ from left to right. Each disk has a black side and white side. Initially all disks...
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John Rhodes
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August 15, 2023
Sum of Squares of Unitary Divisors
A unitary divisor $d$ of a number $n$ is a divisor of $n$ that has the property $\gcd(d, n/d) = 1$. The unitary divisors of $4! =...
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John Rhodes
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August 15, 2023
Necklace of Circles
Let $a$, $b$ and $c$ be positive numbers. Let $W, X, Y, Z$ be four collinear points where $|WX| = a$, $|XY| = b$, $|YZ| = c$...
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John Rhodes
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August 15, 2023
$n$-sequences
A sequence of integers $S = \{s_i\}$ is called an $n$-sequence if it has $n$ elements and each element $s_i$ satisfies $1 \leq s_i \leq n$. Thus...
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John Rhodes
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August 15, 2023
Box-Ball System
Consider an infinite row of boxes. Some of the boxes contain a ball. For example, an initial configuration of 2 consecutive occupied boxes followed by 2 empty...
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John Rhodes
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August 15, 2023
Prime Connection
Two positive numbers $A$ and $B$ are said to be connected (denoted by “$A \leftrightarrow B$”) if one of these conditions holds: (1) $A$ and $B$ have...
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John Rhodes
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August 15, 2023
Kakuro
The above is an example of a cryptic kakuro (also known as cross sums, or even sums cross) puzzle, with its final solution on the right. (The...
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John Rhodes
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August 15, 2023
Consecutive Die Throws
Let $n$ be a positive integer. A 6-sided die is thrown $n$ times. Let $c$ be the number of pairs of consecutive throws that give the same...