A Preference for A5
A printing shop runs 16 batches (jobs) every week and each batch requires a sheet of special colour-proofing paper of size A5. Every Monday morning, the supervisor...
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John Rhodes
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August 15, 2023
Sub-triangle Sums
In a triangular array of positive and negative integers, we wish to find a sub-triangle such that the sum of the numbers it contains is the smallest...
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John Rhodes
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August 15, 2023
Maximum-sum Subsequence
Looking at the table below, it is easy to verify that the maximum possible sum of adjacent numbers in any direction (horizontal, vertical, diagonal or anti-diagonal) is...
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John Rhodes
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August 15, 2023
Exploring Pascal’s Triangle
We can easily verify that none of the entries in the first seven rows of Pascal’s triangle are divisible by $7$: $1$ $1$ $1$ $1$ $2$ $1$...
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John Rhodes
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August 15, 2023
Rectangles in Cross-hatched Grids
In a $3 \times 2$ cross-hatched grid, a total of $37$ different rectangles could be situated within that grid as indicated in the sketch. There are $5$...
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John Rhodes
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August 15, 2023
Investigating a Prime Pattern
The smallest positive integer $n$ for which the numbers $n^2 + 1$, $n^2 + 3$, $n^2 + 7$, $n^2 + 9$, $n^2 + 13$, and $n^2 +...
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John Rhodes
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August 15, 2023
Reversible Numbers
Some positive integers $n$ have the property that the sum $[n + \operatorname{reverse}(n)]$ consists entirely of odd (decimal) digits. For instance, $36 + 63 = 99$ and...
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John Rhodes
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August 15, 2023
Laser Beam Reflections
In laser physics, a “white cell” is a mirror system that acts as a delay line for the laser beam. The beam enters the cell, bounces around...