Solution to: Postman Pat

If postman Pat had delivered mail three times at each house, then the total sum of the house numbers per day would be (1+2+3+4+5+6+7+8+9+10)×3=165. Now that sum is 18+12+23+19+32+25=129. The difference is 165-129=36; divided by 3 this is 12. The sum of the house numbers where no mail was delivered is therefore 12. The following combinations are possible:

2+10
3+9
4+8
5+7

Each day, at four houses the mail was delivered. On Tuesday, the sum was 12. 12 can only be made from four house numbers in two ways:

1+2+3+6
1+2+4+5

The same holds for Friday with the sum of 32:

5+8+9+10
6+7+9+10

From this, we can conclude that the house numbers 1, 2, 9 and 10 for sure have received mail, which means that the combinations 2+10 and 3+9 are not possible. In addition, the combination 5+7 is not possible, because mail was delivered either at house 5 or at house 7. Thus the only remaining solution is: houses 4 and 8.

N.B.: there are various possibilities for the actual post-delivery of the whole week. For example:

 Monday houses 1, 3, 5 and 9 Tuesday houses 1, 2, 3 and 6 Wednesday houses 1, 5, 7 and 10 Thursday houses 2, 3, 5 and 9 Friday houses 6, 7, 9 and 10 Saturday houses 2, 6, 7 and 10