Robot Walks
A robot moves in a series of one-fifth circular arcs ($72^\circ$), with a free choice of a clockwise or an anticlockwise arc for each step, but no...
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John Rhodes
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August 15, 2023
Integer Partition Equations
For some positive integers $k$, there exists an integer partition of the form $4^t = 2^t + k$, where $4^t$, $2^t$, and $k$ are all positive integers...
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John Rhodes
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August 15, 2023
Concealed Square
Find the unique positive integer whose square has the form 1_2_3_4_5_6_7_8_9_0, where each “_” is a single digit. This problem is challenging because it limits the number...
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John Rhodes
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August 15, 2023
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...