Hopping Game
Let $s_k$ be the number of 1’s when writing the numbers from 0 to $k$ in binary. For example, writing 0 to 5 in binary, we have...
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John Rhodes
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August 15, 2023
Triangles with Non Rational Sides and Integral Area
Consider the triangle with sides $\sqrt 5$, $\sqrt {65}$ and $\sqrt {68}$. It can be shown that this triangle has area $9$. $S(n)$ is the sum of...
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John Rhodes
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August 15, 2023
Platonic Dice
An unbiased single $4$-sided die is thrown and its value, $T$, is noted.$T$ unbiased $6$-sided dice are thrown and their scores are added together. The sum, $C$,...
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John Rhodes
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August 15, 2023
Distinct Lines
Consider all lattice points $(a,b,c)$ with $0 \le a,b,c \le N$. From the origin $O(0,0,0)$ all lines are drawn to the other lattice points. Let $D(N)$ be...
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John Rhodes
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August 15, 2023
Harshad Numbers
A Harshad or Niven number is a number that is divisible by the sum of its digits. $201$ is a Harshad number because it is divisible by...
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John Rhodes
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August 15, 2023
Maximum Length of an Antichain
Let $n$ be an integer and $S(n)$ be the set of factors of $n$. A subset $A$ of $S(n)$ is called an antichain of $S(n)$ if $A$...
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John Rhodes
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August 15, 2023
Ellipses Inside Triangles
For any triangle $T$ in the plane, it can be shown that there is a unique ellipse with largest area that is completely inside $T$. For a...
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John Rhodes
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August 15, 2023
Rudin-Shapiro Sequence
Define the sequence $a(n)$ as the number of adjacent pairs of ones in the binary expansion of $n$ (possibly overlapping). E.g.: $a(5) = a(101_2) = 0$, $a(6)...