Dice Game
Peter has nine four-sided (pyramidal) dice, each with faces numbered $1, 2, 3, 4$. Colin has six six-sided (cubic) dice, each with faces numbered $1, 2, 3,...
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John Rhodes
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August 15, 2023
Generalised Hamming Numbers
A Hamming number is a positive number which has no prime factor larger than $5$. So the first few Hamming numbers are $1, 2, 3, 4, 5,...
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John Rhodes
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August 15, 2023
Squarefree Binomial Coefficients
The binomial coefficients $\displaystyle \binom n k$ can be arranged in triangular form, Pascal’s triangle, like this: 111121133114641151010511615201561172135352171 ……… It can be seen that the first eight...
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John Rhodes
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August 15, 2023
Laserbeam
Three mirrors are arranged in the shape of an equilateral triangle, with their reflective surfaces pointing inwards. There is an infinitesimal gap at each vertex of the...
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John Rhodes
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August 15, 2023
Subsets with a Unique Sum
For any set $A$ of numbers, let $\operatorname{sum}(A)$ be the sum of the elements of $A$. Consider the set $B = \{1,3,6,8,10,11\}$. There are $20$ subsets of...
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John Rhodes
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August 15, 2023
Prime-proof Squbes
We shall define a sqube to be a number of the form, $p^2 q^3$, where $p$ and $q$ are distinct primes. For example, $200 = 5^2 2^3$...
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John Rhodes
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August 15, 2023
Iterative Circle Packing
Three circles of equal radius are placed inside a larger circle such that each pair of circles is tangent to one another and the inner circles do...
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John Rhodes
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August 15, 2023
Ambiguous Numbers
A best approximation to a real number $x$ for the denominator bound $d$ is a rational number $\frac r s$ (in reduced form) with $s \le d$,...