Binary Circles
$2^N$ binary digits can be placed in a circle so that all the $N$-digit clockwise subsequences are distinct. For $N=3$, two such circular arrangements are possible, ignoring...
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John Rhodes
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August 15, 2023
Triangle Centres
Consider all the triangles having: All their vertices on lattice pointsInteger coordinates. CircumcentreCentre of the circumscribed circle at the origin $O$. OrthocentrePoint where the three altitudes meet...
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John Rhodes
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August 15, 2023
An Engineers’ Dream Come True
Consider the number $6$. The divisors of $6$ are: $1,2,3$ and $6$. Every number from $1$ up to and including $6$ can be written as a sum...
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John Rhodes
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August 15, 2023
Mountain Range
The following equation represents the continuous topography of a mountainous region, giving the elevationheight above sea level $h$ at any point $(x, y)$: $$h = \left(5000 –...
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John Rhodes
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August 15, 2023
Pivotal Square Sums
Let us call a positive integer $k$ a square-pivot, if there is a pair of integers $m \gt 0$ and $n \ge k$, such that the sum...
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John Rhodes
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August 15, 2023
Stone Game
A game is played with three piles of stones and two players. On each player’s turn, the player may remove one or more stones from the piles....
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John Rhodes
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August 15, 2023
Reachable Numbers
A positive integer will be called reachable if it can result from an arithmetic expression obeying the following rules: Uses the digits $1$ through $9$, in that...
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John Rhodes
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August 15, 2023
A Lagged Fibonacci Sequence
A sequence is defined as: $g_k = 1$, for $0 \le k \le 1999$ $g_k = g_{k-2000} + g_{k – 1999}$, for $k \ge 2000$. Find $g_k...