Primes with Runs
Considering $4$-digit primes containing repeated digits it is clear that they cannot all be the same: $1111$ is divisible by $11$, $2222$ is divisible by $22$, and...
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John Rhodes
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August 15, 2023
Diophantine Reciprocals II
In the following equation $x$, $y$, and $n$ are positive integers. $$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$$ It can be verified that when $n = 1260$ there are...
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John Rhodes
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August 15, 2023
Darts
In the game of darts a player throws three darts at a target board which is split into twenty equal sized sections numbered one to twenty. The...
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John Rhodes
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August 15, 2023
Diophantine Reciprocals I
In the following equation $x$, $y$, and $n$ are positive integers. $$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$$ For $n = 4$ there are exactly three distinct solutions: $$\begin{align}...
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John Rhodes
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August 15, 2023
Minimal Network
The following undirected network consists of seven vertices and twelve edges with a total weight of 243. The same network can be represented by the matrix below....
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John Rhodes
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August 15, 2023
Special Subset Sums: Meta-testing
Let $S(A)$ represent the sum of elements in set $A$ of size $n$. We shall call it a special sum set if for any two non-empty disjoint...
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John Rhodes
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August 15, 2023
Special Subset Sums: Testing
Let $S(A)$ represent the sum of elements in set $A$ of size $n$. We shall call it a special sum set if for any two non-empty disjoint...
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John Rhodes
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August 15, 2023
Pandigital Fibonacci Ends
The Fibonacci sequence is defined by the recurrence relation: $F_n = F_{n – 1} + F_{n – 2}$, where $F_1 = 1$ and $F_2 = 1$. It...