Binary Series
For $0 \le x \lt 1$, define $d_i(x)$ to be the $i$th digit after the binary point of the binary representation of $x$. For example $d_2(0.25) =...
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John Rhodes
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August 16, 2023
Piles of Plates
We stack $n$ plates into $k$ non-empty piles where each pile is a different size. Define $f(n,k)$ to be the maximum number of plates possible in the...
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John Rhodes
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August 16, 2023
Shuffling Cards
A standard deck of $52$ playing cards, which consists of thirteen ranks (Ace, Two, …, Ten, King, Queen and Jack) each in four suits (Clubs, Diamonds, Hearts...
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John Rhodes
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August 16, 2023
Powers of Two
$2^7=128$ is the first power of two whose leading digits are “12”. The next power of two whose leading digits are “12” is $2^{80}$. Define $p(L, n)$...
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John Rhodes
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August 16, 2023
Inverse Digit Sum II
Writing down the numbers which have a digit sum of 10 in ascending order, we get: $19, 28, 37, 46,55,64,73,82,91,109, 118,\dots$ Let $f(n,m)$ be the $m^{\text{th}}$ occurrence...
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John Rhodes
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August 16, 2023
Inverse Digit Sum
Define $s(n)$ to be the smallest number that has a digit sum of $n$. For example $s(10) = 19$. Let $\displaystyle S(k) = \sum_{n=1}^k s(n)$. You are...
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John Rhodes
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August 16, 2023
The Chase II
Consider the following variant of “The Chase” game. This game is played between $n$ players sitting around a circular table using two dice. It consists of $n-1$...
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John Rhodes
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August 16, 2023
$5$-Smooth Pairs
$5$-smooth numbers are numbers whose largest prime factor doesn’t exceed $5$. $5$-smooth numbers are also called Hamming numbers. Let $\Omega(a)$ be the count of prime factors of...