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...
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John Rhodes
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August 15, 2023
Collatz Prefix Families
The Collatz sequence is defined as: $a_{i+1} = \left\{ \large{\frac {a_i} 2 \atop 3 a_i+1} {\text{if }a_i\text{ is even} \atop \text{if }a_i\text{ is odd}} \right.$. The Collatz...
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John Rhodes
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August 15, 2023
Under the Rainbow
$70$ coloured balls are placed in an urn, $10$ for each of the seven rainbow colours. What is the expected number of distinct colours in $20$ randomly...
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John Rhodes
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August 15, 2023
Exploding Sequence
Define the sequence $a_1, a_2, a_3, \dots$ as: $a_1 = 1$ $a_{n+1} = 6a_n^2 + 10a_n + 3$ for $n \ge 1$. Examples: $a_3 = 2359$ $a_6...