Shortest Lattice Vector
Let $t_n$ be the tribonacci numbers defined as: $t_0 = t_1 = 0$; $t_2 = 1$; $t_n = t_{n-1} + t_{n-2} + t_{n-3}$ for $n \ge 3$...
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John Rhodes
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August 15, 2023
Clock Sequence
Consider the infinite repeating sequence of digits: 1234321234321234321… Amazingly, you can break this sequence of digits into a sequence of integers such that the sum of the...
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John Rhodes
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August 15, 2023
Bidirectional Recurrence
Let: $\begin{array}{ll} x(0)&=0 \\ x(1)&=1 \\ x(2k)&=(3x(k)+2x(\lfloor \frac k 2 \rfloor)) \text{ mod } 2^{60} \text{ for } k \ge 1 \text {, where } \lfloor \text...
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John Rhodes
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August 15, 2023
Square on the Inside
Let $ABCD$ be a quadrilateral whose vertices are lattice points lying on the coordinate axes as follows: $A(a, 0)$, $B(0, b)$, $C(-c, 0)$, $D(0, -d)$, where $1...
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John Rhodes
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August 15, 2023
Compromise or Persist
Alice is playing a game with $n$ cards numbered $1$ to $n$. A game consists of iterations of the following steps. (1) Alice picks one of the...
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John Rhodes
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August 15, 2023
Counting Castles
We define a block to be a rectangle with a height of $1$ and an integer-valued length. Let a castle be a configuration of stacked blocks. Given...
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John Rhodes
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August 15, 2023
Eight Divisors
The eight divisors of $24$ are $1, 2, 3, 4, 6, 8, 12$ and $24$. The ten numbers not exceeding $100$ having exactly eight divisors are $24,...
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John Rhodes
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August 15, 2023
Problem 500!!!
The number of divisors of $120$ is $16$. In fact $120$ is the smallest number having $16$ divisors. Find the smallest number with $2^{500500}$ divisors. Give your...