Squares on the Line
Sam and Tom are trying a game of (partially) covering a given line segment of length $L$ by taking turns in placing unit squares onto the line...
-
John Rhodes
-
August 16, 2023
$2$-Friendly
Two positive integers $a$ and $b$ are $2$-friendly when $\gcd(a,b) = 2^t, t \gt 0$. For example, $24$ and $40$ are $2$-friendly because $\gcd(24,40) = 8 =...
-
John Rhodes
-
August 16, 2023
A Long Row of Dice
Consider a row of $n$ dice all showing 1. First turn every second die,$ (2,4,6,\ldots)$, so that the number showing is increased by 1. Then turn every...
-
John Rhodes
-
August 16, 2023
Shut the Box
Bob plays a single-player game of chance using two standard 6-sided dice and twelve cards numbered 1 to 12. When the game starts, all cards are placed...
-
John Rhodes
-
August 16, 2023
Summing a Multiplicative Function
A multiplicative function $f(x)$ is a function over positive integers satisfying $f(1)=1$ and $f(a b)=f(a) f(b)$ for any two coprime positive integers $a$ and $b$. For integer...
-
John Rhodes
-
August 16, 2023
Weighted Lattice Paths
Let $P_{a,b}$ denote a path in a $a\times b$ lattice grid with following properties: The path begins at $(0,0)$ and ends at $(a,b)$. The path consists only...
-
John Rhodes
-
August 16, 2023
Flexible Digit Sum
Given any positive integer $n$, we can construct a new integer by inserting plus signs between some of the digits of the base $B$ representation of $n$,...
-
John Rhodes
-
August 16, 2023
Subset Sums
Let $A_q(n)$ be the number of subsets, $B$, of the set $\{1, 2, …, q \cdot n\}$ that satisfy two conditions: 1) $B$ has exactly $n$ elements;...