Strong Repunits
The number $7$ is special, because $7$ is $111$ written in base $2$, and $11$ written in base $6$ (i.e. $7_{10} = 11_6 = 111_2$). In other...
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John Rhodes
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August 15, 2023
Matrix Sum
We define the Matrix Sum of a matrix as the maximum possible sum of matrix elements such that none of the selected elements share the same row...
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John Rhodes
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August 15, 2023
Silver Dollar Game
One variant of N.G. de Bruijn’s silver dollar game can be described as follows: On a strip of squares a number of coins are placed, at most...
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John Rhodes
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August 15, 2023
Fractional Sequences
For any positive integer $k$, a finite sequence $a_i$ of fractions $x_i/y_i$ is defined by: $a_1 = 1/k$ and $a_i = (x_{i – 1} + 1) /...
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John Rhodes
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August 15, 2023
The Totient of a Square Is a Cube
Consider the number $50$. $50^2 = 2500 = 2^2 \times 5^4$, so $\phi(2500) = 2 \times 4 \times 5^3 = 8 \times 5^3 = 2^3 \times 5^3$....
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John Rhodes
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August 15, 2023
Golomb’s Self-describing Sequence
The Golomb’s self-describing sequence $(G(n))$ is the only nondecreasing sequence of natural numbers such that $n$ appears exactly $G(n)$ times in the sequence. The values of $G(n)$...
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John Rhodes
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August 15, 2023
Crazy Function
For fixed integers $a, b, c$, define the crazy function $F(n)$ as follows: $F(n) = n – c$ for all $n \gt b$ $F(n) = F(a +...
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John Rhodes
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August 15, 2023
Peredur Fab Efrawg
“And he came towards a valley, through which ran a river; and the borders of the valley were wooded, and on each side of the river were...