Happy New Year!
Earlier, while drinking from my 1729 Maths School mug and thinking about the new year, I noticed something interesting – if you separate 2026 into two 2-digit numbers – 20 and 26 – they have the same mean as 17 and 29.
And I wondered – is this common?
How many other positive 4-digit numbers abcd (with a≠0) can be separated into two 2-digit numbers ab and cd with the same mean as 2026?
And then I wondered a bit more. What if we separate 2026 into two 3-digit numbers 202 and 026 (which we can interpret as 26).
How many other positive 4-digit numbers abcd (with a≠0) can be separated into two 3-digit numbers abc and bcd with the same mean as 2026?
Finally, I wondered what would happen if we just separate 2026 into its individual digits 2, 0, 2 and 6.
How many other positive 4-digit numbers abcd (with a≠0) are there whose digits have the same mean as the digits of 2026?