Three Consecutive Digital Sum Limit
How many $20$ digit numbers $n$ (without any leading zero) exist such that no three consecutive digits of $n$ have a sum greater than $9$? To generate...
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John Rhodes
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August 15, 2023
Cross-hatched Triangles
Consider an equilateral triangle in which straight lines are drawn from each vertex to the middle of the opposite side, such as in the size $1$ triangle...
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John Rhodes
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August 15, 2023
Hexadecimal Numbers
In the hexadecimal number system numbers are represented using $16$ different digits: $$0,1,2,3,4,5,6,7,8,9,\mathrm A,\mathrm B,\mathrm C,\mathrm D,\mathrm E,\mathrm F.$$ The hexadecimal number $\mathrm{AF}$ when written in the...
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John Rhodes
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August 15, 2023
Triominoes
A triomino is a shape consisting of three squares joined via the edges. There are two basic forms: If all possible orientations are taken into account there...
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John Rhodes
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August 15, 2023
Factorial Trailing Digits
For any $N$, let $f(N)$ be the last five digits before the trailing zeroes in $N!$. For example, $9! = 362880$ so $f(9)=36288$ $10! = 3628800$ so...
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John Rhodes
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August 15, 2023
Digital Root Sums of Factorisations
A composite number can be factored many different ways. For instance, not including multiplication by one, $24$ can be factored in $7$ distinct ways: \begin{align} 24 &=...
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John Rhodes
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August 15, 2023
Lexicographical Neighbours
Taking three different letters from the $26$ letters of the alphabet, character strings of length three can be formed. Examples are ‘abc’, ‘hat’ and ‘zyx’. When we...
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John Rhodes
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August 15, 2023
Base-10 Diophantine Reciprocal
Consider the diophantine equation $\frac 1 a + \frac 1 b = \frac p {10^n}$ with $a, b, p, n$ positive integers and $a \le b$. For...