Postman Nat delivers the mail in a small village which has only one street with exactly ten houses, numbered from 1 up to and including 10.

In a certain week, Nat did not deliver any mail at two houses in the village; at the other houses he delivered mail three times each. Each working day he delivered mail at exactly four houses.

The sums of the house numbers where he delivered mail were:

on Monday: 18
on Tuesday: 12
on Wednesday: 23
on Thursday: 19
on Friday: 32
op Saturday: 25
on Sunday: he never works

Which two houses didn't get any mail that week?

a) The total for the six days is 129. Dividing by three gives the total of the numbers of the houses receiving mail: 43

b) The total of all the house numbers is 55, so by subtraction the total of the two not receiving mail is 12.

c) The two not receiving mail must be [2,10] or [3,9] or [4,8] or [5,7]

d) So who received mail on Friday, when the total was 32?

10 and 9 both received mail on that day, because without both of them there are not enough house numbers to add up to 32.

The other two who did receive mail must be either [8,5] or [7,6]

But it cannot be 10 and 9 and 8 and 5 who received Friday mail, because that rules out all 4 of the possible sets of non-recipients in (c).

So Friday mail was received by 10 and 9 and 7 and 6.

e) So the houses who did not get any mail all week were 4 and 8. Apparently, nobody loves a multiple of 4.

*Edited on ***June 11, 2013, 4:54 pm**