by Serhiy Grabarchuk Jr
Alison’s phone number consists of all 10 digits. No repeats. On top of that the sum of two neighboring digits is always either 9 or 10.
What is the first digit in Alison’s phone number if it is not 0?
Each digit in the phone number, except for the first and the last one, has a left and a right neighbor, producing either the sum of 9 or 10 with them. The 0 can only have one neighbor 9, producing the sum of 9 with it. The 5 can only have one neighbor 4, producing the sum of 9 with it. Thus, 0 and 5 are on the two opposite ends of the phone number. Knowing that 0 is not at the beginning, it is exactly 5 which is the first digit.
BTW, knowing the first digit it is possible to discover the entire Alison’s phone number.