Reciprocal Games I
Tom has built a random generator that is connected to a row of $n$ light bulbs. Whenever the random generator is activated each of the $n$ lights...
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John Rhodes
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August 15, 2023
Cake Icing Puzzle
Adam plays the following game with his birthday cake. He cuts a piece forming a circular sector of $60$ degrees and flips the piece upside down, with...
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John Rhodes
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August 15, 2023
Divisibility of Sum of Divisors
Let $\sigma(n)$ be the sum of the divisors of $n$. E.g. the divisors of $4$ are $1$, $2$ and $4$, so $\sigma(4)=7$. The numbers $n$ not exceeding...
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John Rhodes
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August 15, 2023
Maximal Polygons
A line segment of length $2n-3$ is randomly split into $n$ segments of integer length ($n \ge 3$). In the sequence given by this split, the segments...
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John Rhodes
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August 15, 2023
Robot Welders
A company specialises in producing large rectangular metal sheets, starting from unit square metal plates. The welding is performed by a range of robots of increasing size....
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John Rhodes
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August 15, 2023
Maximal Perimeter
Construct triangle $ABC$ such that: Vertices $A$, $B$ and $C$ are lattice points inside or on the circle of radius $r$ centered at the origin; the triangle...
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John Rhodes
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August 15, 2023
Divisor Pairs
Let $S(n)$ be the number of pairs $(a,b)$ of distinct divisors of $n$ such that $a$ divides $b$. For $n=6$ we get the following pairs: $(1,2), (1,3),...
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John Rhodes
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August 15, 2023
Coprime Nim
Coprime Nim is just like ordinary normal play Nim, but the players may only remove a number of stones from a pile that is coprime with the...