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...
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John Rhodes
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August 15, 2023
St. Petersburg Lottery
A gambler decides to participate in a special lottery. In this lottery the gambler plays a series of one or more games. Each game costs $m$ pounds...
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John Rhodes
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August 15, 2023
Remainder of Polynomial Division
For positive integers $n$ and $m$, we define two polynomials $F_n(x) = x^n$ and $G_m(x) = (x-1)^m$. We also define a polynomial $R_{n,m}(x)$ as the remainder of...
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John Rhodes
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August 15, 2023
Drunken Tower of Hanoi
Bob is very familiar with the famous mathematical puzzle/game, “Tower of Hanoi,” which consists of three upright rods and disks of different sizes that can slide onto...
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John Rhodes
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August 15, 2023
Incenter and Circumcenter of Triangle
Given an integer sided triangle $ABC$: Let $I$ be the incenter of $ABC$. Let $D$ be the intersection between the line $AI$ and the circumcircle of $ABC$...
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John Rhodes
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August 15, 2023
Writing $n$ as the Product of $k$ Distinct Positive Integers
Let $W(n,k)$ be the number of ways in which $n$ can be written as the product of $k$ distinct positive integers. For example, $W(144,4) = 7$. There...