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...
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John Rhodes
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August 16, 2023
Maximal Area
Given positive integers $a \le b \le c \le d$, it may be possible to form quadrilaterals with edge lengths $a,b,c,d$ (in any order). When this is...
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John Rhodes
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August 16, 2023
Yarra Gnisrever
Let $N$ and $K$ be two positive integers. $F_n$ is the $n$-th Fibonacci number: $F_1 = F_2 = 1$, $F_n = F_{n – 1} + F_{n –...
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John Rhodes
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August 16, 2023
Freefarea
Let $S$ be the set consisting of the four letters $\{\texttt{`A’},\texttt{`E’},\texttt{`F’},\texttt{`R’}\}$. For $n\ge 0$, let $S^*(n)$ denote the set of words of length $n$ consisting of letters...