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Published in Brain Teasers

Madadia is split, like other countries, into cities, towns and villages. Some are large and some are small. Postman Patel delivers the mail in the village of "Lardville".

Now Lardville is so small there is only one street with only ten houses, numbered from 1 to 10.

In a certain week, Patel 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: His day off

Can you work out which two houses received no mail this week?

Solution:

If postman Patel would have 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)x3=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 2 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. Also 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.

Today's brain teaser courtesy of Braingle.com.