Few Repeated Digits
How many $18$-digit numbers $n$ (without leading zeros) are there such that no digit occurs more than three times in $n$? This is a rather complex combinatorics...
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John Rhodes
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August 15, 2023
Square Sum of the Digital Squares
For a positive integer $n$, let $f(n)$ be the sum of the squares of the digits (in base $10$) of $n$, e.g. \begin{align} f(3) &= 3^2 =...
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John Rhodes
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August 15, 2023
Pandigital Concatenating Products
Take the number $6$ and multiply it by each of $1273$ and $9854$: \begin{align} 6 \times 1273 &= 7638\\ 6 \times 9854 &= 59124 \end{align} By concatenating...
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John Rhodes
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August 15, 2023
Sums of Powers of Two
Define $f(0)=1$ and $f(n)$ to be the number of different ways $n$ can be expressed as a sum of integer powers of $2$ using each power no...
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John Rhodes
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August 15, 2023
Number Rotations
Consider the number $142857$. We can right-rotate this number by moving the last digit ($7$) to the front of it, giving us $714285$. It can be verified...
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John Rhodes
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August 15, 2023
Investigating Ulam Sequences
For two positive integers $a$ and $b$, the Ulam sequence $U(a,b)$ is defined by $U(a,b)_1 = a$, $U(a,b)_2 = b$ and for $k \gt 2$, $U(a,b)_k$ is...
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John Rhodes
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August 15, 2023
Criss Cross
A $4 \times 4$ grid is filled with digits $d$, $0 \le d \le 9$. It can be seen that in the grid \begin{matrix} 6 & 3...
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John Rhodes
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August 15, 2023
Intersections
A segment is uniquely defined by its two endpoints. By considering two line segments in plane geometry there are three possibilities: the segments have zero points, one...