Lattice Points on a Circle
Let $f(N)$ be the number of points with integer coordinates that are on a circle passing through $(0,0)$, $(N,0)$,$(0,N)$, and $(N,N)$. It can be shown that $f(10000)...
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John Rhodes
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August 15, 2023
The Race
Two players share an unbiased coin and take it in turns to play The Race. On Player 1’s turn, the coin is tossed once. If it comes...
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John Rhodes
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August 15, 2023
Prime Factorisation of Binomial Coefficients
The binomial coefficient $\displaystyle \binom {10} 3 = 120$. $120 = 2^3 \times 3 \times 5 = 2 \times 2 \times 2 \times 3 \times 5$, and...
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John Rhodes
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August 15, 2023
Fibonacci Words
For any two strings of digits, $A$ and $B$, we define $F_{A, B}$ to be the sequence $(A,B,AB,BAB,ABBAB,\dots)$ in which each term is the concatenation of the...
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John Rhodes
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August 15, 2023
Four Representations Using Squares
Consider the number $3600$. It is very special, because \begin{alignat}{2} 3600 &= 48^2 + &&36^2\\ 3600 &= 20^2 + 2 \times &&40^2\\ 3600 &= 30^2 + 3...
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John Rhodes
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August 15, 2023
Minkowski Sums
Let $S_n$ be the regular $n$-sided polygon – or shape – whose vertices $v_k$ ($k = 1, 2, \dots, n$) have coordinates: \begin{align} x_k &= \cos((2k –...
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John Rhodes
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August 15, 2023
The Chase
The Chase is a game played with two dice and an even number of players. The players sit around a table and the game begins with two...
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John Rhodes
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August 15, 2023
Tribonacci Non-divisors
The sequence $1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355, 653, 1201, \dots$ is defined by $T_1 = T_2 = T_3 = 1$...