Fractions and Sum of Powers of Two
Define $f(0)=1$ and $f(n)$ to be the number of ways to write $n$ as a sum of powers of $2$ where no power occurs more than twice....
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John Rhodes
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August 15, 2023
Hollow Square Laminae II
We shall define a square lamina to be a square outline with a square “hole” so that the shape possesses vertical and horizontal symmetry. Given eight tiles...
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John Rhodes
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August 15, 2023
Hollow Square Laminae I
We shall define a square lamina to be a square outline with a square “hole” so that the shape possesses vertical and horizontal symmetry. For example, using...
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John Rhodes
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August 15, 2023
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...