Wiping out a number from 1 to 6 and leaving behind an average which is a whole number is possible if you wipe out the number 1. Wiping out 1 leaves you with 2,3,4,5,6 which adds up to a total of 20. 20 divided by the number of numbers (5) gives you 4 which is a whole number.
It is also possible by wiping out the number 6. Wiping out 6 leaves you with 1,2,3,4,5 which adds up to a total of 15. 15 divided by the number of numbers (5) gives you 3 which is a whole number.
What about starting with other sets of numbers from 1 to N where N is even, wiping out just one number, and finding the mean?
Which numbers can be wiped out so that the mean of what is left is a whole number?
1+2+3+4+5+6+7+8=36
This time if you remove a number then for the mean to be a whole number the remaining sum needs to be a multiple of 7.
You can take away the numbers 8 and 1 to get a multiple of 7 (28 and 35).
How about 1+2+...+9+10?
The answer is 10 and 1.
1+2+....+9+10=55. If you remove a number then the remaining sum needs to be a multiple of 9.
What about 1+2+...+N where N is an even number?
My theory is that removing 1 and N would make the mean of the remaining N-1 numbers a whole number.
Now let's try odd N starting with 1+2+3+4+5.
1+2+3+4+5=15
You can take away the number 3 from 15 because it gives you 12 which is a multiple of 4.
The next multiple of 4 after 12 is 8 and you cannot take away any of the numbers from 1 to 5 from 15 to give you 8. So 3 is the only number that you can take away to make the mean of the remaining numbers a whole number.
1+2+...+6+7=28
You can take away the number 4 from 28 because it gives 24 which is a multiple of 6.
1+2+....+8+9=45
You can take away the number 5 from 45 because it gives 40 which is a multiple of 8.
1+2+....+10+11=66
You can take away the number 6 from 66 because it gives 60 which is a multiple of 10.
Is there a pattern?
The answers so far have been: 3,4,5,6
How do you get from 5 to 3, 7 to 4, 9 to 5 and 11 to 6?
You add 1 and divide by 2.
So if I had 1+2+....+99, I would take away 50 to make the mean of the remaining numbers a whole number.
Puzzling Wipeouts 1:
The number which was crossed out was 3.
1+2+4+5+6 = 18 and 18 divided by 5 is 3 remainder 3 or 3.6.
Puzzling Wipeouts 2:
The number which was crossed out was 12.
1+2+....+10+11+13+14+15 = 108 and 108 divided by 14 = 7.714285.