Almost Right-angled Triangles II
Let us call an integer sided triangle with sides $a \le b \le c$ barely obtuse if the sides satisfy $a^2 + b^2 = c^2 – 1$....
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John Rhodes
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August 15, 2023
Almost Right-angled Triangles I
Let us call an integer sided triangle with sides $a \le b \le c$ barely acute if the sides satisfy $a^2 + b^2 = c^2 + 1$....
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John Rhodes
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August 15, 2023
Sphere Packing
What is the length of the shortest pipe, of internal radius $\pu{50 mm}$, that can fully contain $21$ balls of radii $\pu{30 mm}, \pu{31 mm}, \dots, \pu{50...
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John Rhodes
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August 15, 2023
Alexandrian Integers
We shall call a positive integer $A$ an “Alexandrian integer”, if there exist integers $p, q, r$ such that: $$A = p \cdot q \cdot r$$ and...
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John Rhodes
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August 15, 2023
Heighway Dragon
Let D0 be the two-letter string “Fa”. For n≥1, derive Dn from Dn-1 by the string-rewriting rules: “a” → “aRbFR” “b” → “LFaLb” Thus, D0 = “Fa”,...
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John Rhodes
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August 15, 2023
Skew-cost Coding
Let A and B be bit strings (sequences of 0’s and 1’s). If A is equal to the leftmost length(A) bits of B, then A is said...
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John Rhodes
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August 15, 2023
Perfect Right-angled Triangles
Consider the right angled triangle with sides $a=7$, $b=24$ and $c=25$. The area of this triangle is $84$, which is divisible by the perfect numbers $6$ and...
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John Rhodes
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August 15, 2023
Balanced Numbers
A positive integer with $k$ (decimal) digits is called balanced if its first $\lceil k/2 \rceil$ digits sum to the same value as its last $\lceil k/2...