17. Students in a class take turns to practise their arithmetic skills. Initially a board contains the integers from 1 to 10 inclusive, each written ten times. On each turn a student first deletes two of the integers and then writes on the board the number that is one more than the sum of those two deleted integers. Turns are taken until there is only one number remaining on the board. Assuming no student makes a mistake, what is the remaining number?
17. 649 We observe that the total of all integers on the board at the start of the process is $10(1+2+3+4+5+6+7+8+9+10)=550$.
On each turn this total is increased by 1 . Since we start with one hundred integers on the board and at each turn this number of integers is decreased by one, then 99 turns will be required to complete the process. Therefore the total of all integers on the board will increase by 99 over the course of the process. Hence the remaining number will be $550+99=649$.