Digit Factorial Chains
The number $145$ is well known for the property that the sum of the factorial of its digits is equal to $145$: $$1! + 4! + 5!...
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John Rhodes
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August 15, 2023
Counting Fractions in a Range
Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n, d)=1$, it is called a reduced proper fraction....
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John Rhodes
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August 15, 2023
Counting Fractions
Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n,d)=1$, it is called a reduced proper fraction. If...
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John Rhodes
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August 15, 2023
Ordered Fractions
Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n,d)=1$, it is called a reduced proper fraction. If...
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John Rhodes
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August 15, 2023
Totient Permutation
Euler’s totient function, $\phi(n)$ [sometimes called the phi function], is used to determine the number of positive numbers less than or equal to $n$ which are relatively...
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John Rhodes
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August 15, 2023
Totient Maximum
Euler’s totient function, $\phi(n)$ [sometimes called the phi function], is defined as the number of positive integers not exceeding $n$ which are relatively prime to $n$. For...
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John Rhodes
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August 15, 2023
Magic 5-gon Ring
Consider the following “magic” 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine. Working clockwise, and starting from the group of...
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John Rhodes
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August 15, 2023
Maximum Path Sum II
By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23....