Twos Are All You Need
A positive integer, $n$, is factorised into prime factors. We define $f(n)$ to be the product when each prime factor is replaced with $2$. In addition we...
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John Rhodes
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August 16, 2023
$3$-Like Numbers
For a positive integer $n$, define $f(n)$ to be the number of non-empty substrings of $n$ that are divisible by $3$. For example, the string “2573” has...
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John Rhodes
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August 16, 2023
Total Inversion Count of Divided Sequences
The inversion count of a sequence of digits is the smallest number of adjacent pairs that must be swapped to sort the sequence. For example, $34214$ has...
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John Rhodes
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August 16, 2023
Factors of Two in Binomial Coefficients
Define $g(n, m)$ to be the largest integer $k$ such that $2^k$ divides $\binom{n}m$. For example, $\binom{12}5 = 792 = 2^3 \cdot 3^2 \cdot 11$, hence $g(12,...
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John Rhodes
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August 16, 2023
Circular Logic II
Given an integer $n$, $n \geq 3$, let $B=\{\mathrm{false},\mathrm{true}\}$ and let $B^n$ be the set of sequences of $n$ values from $B$. The function $f$ from $B^n$...
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John Rhodes
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August 16, 2023
Random Connected Area
Consider a rectangle made up of $W \times H$ square cells each with area $1$. Each cell is independently coloured black with probability $0.5$ otherwise white. Black...
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John Rhodes
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August 16, 2023
Eulercoin
Leonhard Euler was born on 15 April 1707. Consider the sequence 1504170715041707n mod 4503599627370517. An element of this sequence is defined to be an Eulercoin if it...
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John Rhodes
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August 16, 2023
Triffle Numbers
Let $\sigma(n)$ be the sum of all the divisors of the positive integer $n$, for example: $\sigma(10) = 1+2+5+10 = 18$. Define $T(N)$ to be the sum...