Amicable Chains
The proper divisors of a number are all the divisors excluding the number itself. For example, the proper divisors of $28$ are $1$, $2$, $4$, $7$, and...
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John Rhodes
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August 15, 2023
Almost Equilateral Triangles
It is easily proved that no equilateral triangle exists with integral length sides and integral area. However, the almost equilateral triangle $5$-$5$-$6$ has an area of $12$...
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John Rhodes
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August 15, 2023
Arithmetic Expressions
By using each of the digits from the set, $\{1, 2, 3, 4\}$, exactly once, and making use of the four arithmetic operations ($+, -, \times, /$)...
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John Rhodes
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August 15, 2023
Square Digit Chains
A number chain is created by continuously adding the square of the digits in a number to form a new number until it has been seen before....
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John Rhodes
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August 15, 2023
Right Triangles with Integer Coordinates
The points $P(x_1, y_1)$ and $Q(x_2, y_2)$ are plotted at integer co-ordinates and are joined to the origin, $O(0,0)$, to form $\triangle OPQ$. There are exactly fourteen...
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John Rhodes
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August 15, 2023
Cube Digit Pairs
Each of the six faces on a cube has a different digit ($0$ to $9$) written on it; the same is done to a second cube. By...
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John Rhodes
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August 15, 2023
Roman Numerals
For a number written in Roman numerals to be considered valid there are basic rules which must be followed. Even though the rules allow some numbers to...
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John Rhodes
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August 15, 2023
Product-sum Numbers
A natural number, $N$, that can be written as the sum and product of a given set of at least two natural numbers, $\{a_1, a_2, \dots, a_k\}$...