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...
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John Rhodes
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August 15, 2023
Torricelli Triangles
Let $ABC$ be a triangle with all interior angles being less than $120$ degrees. Let $X$ be any point inside the triangle and let $XA = p$,...
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John Rhodes
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August 15, 2023
Perfect Square Collection
Find the smallest $x + y + z$ with integers $x \gt y \gt z \gt 0$ such that $x + y$, $x – y$, $x +...
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John Rhodes
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August 15, 2023
Square Progressive Numbers
A positive integer, $n$, is divided by $d$ and the quotient and remainder are $q$ and $r$ respectively. In addition $d$, $q$, and $r$ are consecutive positive...