Turán’s Water Heating System
Turan has the electrical water heating system outside his house in a shed. The electrical system uses two fuses in series, one in the house and one...
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John Rhodes
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August 16, 2023
Exponent Difference
For any integer $n>0$ and prime number $p,$ define $\nu_p(n)$ as the greatest integer $r$ such that $p^r$ divides $n$. Define $$D(n, m) = \sum_{p \text{ prime}}...
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John Rhodes
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August 16, 2023
Binary Blackboard
Oscar and Eric play the following game. First, they agree on a positive integer $n$, and they begin by writing its binary representation on a blackboard. They...
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John Rhodes
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August 16, 2023
One Million Members
On Sunday 5 April 2020 the Project Euler membership first exceeded one million members. We would like to present this problem to celebrate that milestone. Thank you...
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John Rhodes
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August 16, 2023
Even Stevens
Every day for the past $n$ days Even Stevens brings home his groceries in a plastic bag. He stores these plastic bags in a cupboard. He either...
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John Rhodes
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August 16, 2023
Twos Are All You Need
A positive integer, $n$, is factorised into prime factors. We define $f(n)$ to be the product when each prime factor is replaced with $2$. In addition we...
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John Rhodes
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August 16, 2023
$3$-Like Numbers
For a positive integer $n$, define $f(n)$ to be the number of non-empty substrings of $n$ that are divisible by $3$. For example, the string “2573” has...
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John Rhodes
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August 16, 2023
Total Inversion Count of Divided Sequences
The inversion count of a sequence of digits is the smallest number of adjacent pairs that must be swapped to sort the sequence. For example, $34214$ has...