ABC Hits
The radical of $n$, $\operatorname{rad}(n)$, is the product of distinct prime factors of $n$. For example, $504 = 2^3 \times 3^2 \times 7$, so $\operatorname{rad}(504) = 2...
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John Rhodes
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August 15, 2023
Cuboid Layers
The minimum number of cubes to cover every visible face on a cuboid measuring $3 \times 2 \times 1$ is twenty-two. If we then add a second...
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John Rhodes
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August 15, 2023
Palindromic Sums
The palindromic number $595$ is interesting because it can be written as the sum of consecutive squares: $6^2 + 7^2 + 8^2 + 9^2 + 10^2 +...
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John Rhodes
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August 15, 2023
Ordered Radicals
The radical of $n$, $\operatorname{rad}(n)$, is the product of the distinct prime factors of $n$. For example, $504 = 2^3 \times 3^2 \times 7$, so $\operatorname{rad}(504) =...
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John Rhodes
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August 15, 2023
Prime Square Remainders
Let $p_n$ be the $n$th prime: $2, 3, 5, 7, 11, \dots$, and let $r$ be the remainder when $(p_n – 1)^n + (p_n + 1)^n$ is...
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John Rhodes
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August 15, 2023
Efficient Exponentiation
The most naive way of computing $n^{15}$ requires fourteen multiplications: $$n \times n \times \cdots \times n = n^{15}.$$ But using a “binary” method you can compute...
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John Rhodes
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August 15, 2023
Disc Game Prize Fund
A bag contains one red disc and one blue disc. In a game of chance a player takes a disc at random and its colour is noted....
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John Rhodes
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August 15, 2023
Square Remainders
Let $r$ be the remainder when $(a – 1)^n + (a + 1)^n$ is divided by $a^2$. For example, if $a = 7$ and $n = 3$,...