Question

In a box there are 21 identical pieces that have been enumerated using odd numbers from 5 to 45. How many tiles should be drawn at random to obtain with certainty 3 tiles whose numbering sum exactly 33?

Answer 1

These are the only combinations of exactly 3 tiles that add to 33.

5,7,21

5,9,19

5,11,17

5,13,15

7,9,17

7,11,15

9,11,13

All tiles with numbers above 21 do not help you. 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45. There are 12 of them.

Every three-number combination must have one of the numbers 5, 7, or 9, so if the six numbers from 13 to 21 are picked, in addition to the 12 higher numbers mentioned above, you already picked 18 tiles, and you still have no solution. To obtain the solutions 5,7,21; 5,9,19; 7,9,17; he needs two more numbers in addition to the 18 he already has, so he needs 20 tiles in total to be guaranteed three of them add to exactly 33.

Answer: 20 tiles