Cutting Rectangular Grid Paper
A rectangular sheet of grid paper with integer dimensions $w \times h$ is given. Its grid spacing is $1$. When we cut the sheet along the grid...
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John Rhodes
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August 15, 2023
Totient Stairstep Sequences
Let $\{a_1, a_2, \dots, a_n\}$ be an integer sequence of length $n$ such that: $a_1 = 6$ for all $1 \le i \lt n$: $\phi(a_i) \lt \phi(a_{i...
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John Rhodes
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August 15, 2023
Maximix Arrangements
A train is used to transport four carriages in the order: ABCD. However, sometimes when the train arrives to collect the carriages they are not in the...
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John Rhodes
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August 15, 2023
Gathering the Beans
Whenever Peter feels bored, he places some bowls, containing one bean each, in a circle. After this, he takes all the beans out of a certain bowl...
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John Rhodes
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August 15, 2023
Spilling the Beans
In Plato’s heaven, there exist an infinite number of bowls in a straight line. Each bowl either contains some or none of a finite number of beans....
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John Rhodes
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August 15, 2023
Special Partitions
All positive integers can be partitioned in such a way that each and every term of the partition can be expressed as $2^i \times 3^j$, where $i,j...
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John Rhodes
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August 15, 2023
Cross Flips
$N \times N$ disks are placed on a square game board. Each disk has a black side and white side. At each turn, you may choose a...
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John Rhodes
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August 15, 2023
Euler’s Number
An infinite sequence of real numbers $a(n)$ is defined for all integers $n$ as follows: $$a(n) = \begin{cases} 1 & n \lt 0\\ \sum \limits_{i = 1}^{\infty}{\dfrac{a(n...