This challenge connects various calculations you can do with the digits that make up an integer. There are plenty of things to explain!

a        In a list of the integers from 1 to 1000000, are there more digits equal to 1, or are there more digits equal to 2?

The digit sum of a number is the total obtained by adding the digits of that number. For example, the digit sum of 3641 is 3 + 6 + 4 + 1 = 14.

If you take the sum of the digits of a number, then the sum of the digits of that number, and so on, you will eventually reach a single digit, called the digital root of the original number. For example, 9785 has digit sum 9 + 7 + 8 + 5 = 29, which has digit sum 2 + 9 = 11, which has digit sum 1 + 1 = 2, so 2 is the digital root of 9785.

b       If each of the integers from 1 to 1000000 is now replaced by its digital root, will there now be more digits equal to 1, or more digits equal to 2 in the list?

c        Prove that the remainder when you divide a number by 9 is the same as the remainder when you divide its digit sum by 9.

d       If you solved part b of this problem by observing a pattern in the digital roots, try to prove what you observed using part c.

The digital persistence of a number is the number of digit sums you have to do before reaching the digital root. For example, the digital persistence of 9785 is 3, because finding the digital root required three digit sums: 9 + 7 + 8 + 5 = 29, 2 + 9 = 11 and 1 + 1 = 2.

e        What is the smallest number with digital persistence 1000000?

The digital difference of a positive integer is the sum of the digits in odd positions (counting from the units digit) minus the sum of the digits in even positions; for a negative integer, the digital difference is the sum of the digits in even positions minus the sum of the digits in odd positions.

For example, the digital difference of 29181 is (2 + 1 + 1) – (8 + 9) = – 13; and the digital difference of –937184 is (9 + 7 + 8) – (3 + 1 + 4) = 16.

The digital reduction of an integer is the result of taking the digital difference repeatedly, until only a single digit (positive or negative) remains.

f           Investigate the pattern in digital reductions. Can you explain any results you observe?

The weekly maths challenge will be back in 2020 - happy holidays!