0809.2: Solution


Solution to Puzzle #2 (November 2008)

Congratulations to Alexander Wood '12, who presented a winning solution.

Other correct solutions were submitted by:

Martin Connor '09 and Sam Brookfield '09; Ari Abrams-Kudan '10; Alexa Ashworth '09; Robert Kosar '12; Thomas Helmuth '09; Yuqi Mao '09; Liz Farrington '10; and

Shoshana Brassfield, Philosophy; Jim Schreve, Physics; Pat Marino, Residential Life

There were a few correct solutions supported by insufficient reasons.

The next puzzle will appear at the very beginning of the spring semester.

Have a Happy Thanksgiving!

 

Drive times and dorm room numbers are positive whole numbers. There are 38 combinations of different positive whole numbers whose sum is less than 18.

        sum product
1 2 3 4 10 24
1 2 3 5 11 30
1 2 3 6 12 36
1 2 3 7 13 42
1 2 3 8 14 48
1 2 3 9 15 54
1 2 3 10 16 60
1 2 3 11 17 66
1 2 4 5 12 40
1 2 4 6 13 48
1 2 4 7 14 56
1 2 4 8 15 64
1 2 4 9 16 72
1 2 4 10 17 80
1 2 5 6 14 60
1 2 5 7 15 70
1 2 5 8 16 80
1 2 5 9 17 90
1 2 6 7 16 84
1 2 6 8 17 96
1 3 4 5 13 60
1 3 4 6 14 72
1 3 4 7 15 84
1 3 4 8 16 96
1 3 4 9 17 108
1 3 5 6 15 90
1 3 5 7 16 105
1 3 5 8 17 120
1 3 6 7 17 126
1 4 5 6 16 120
1 4 5 7 17 140
2 3 4 5 14 120
2 3 4 6 15 144
2 3 4 7 16 168
2 3 4 8 17 192
2 3 5 6 16 180
2 3 5 7 17 210
2 4 5 6 17 240
           

Willard knew the product of the four numbers, since he had come to the women's dorm room. (Anyway, he could easily look.) Since Willard could not determine the numbers from the information given initially, we know that there must be at least two different combinations with the same product. Here is the same table, sorted by the product of the four numbers.

        sum product
1 2 3 4 10 24
1 2 3 5 11 30
1 2 3 6 12 36
1 2 4 5 12 40
1 2 3 7 13 42
1 2 3 8 14 48
1 2 4 6 13 48
1 2 3 9 15 54
1 2 4 7 14 56
1 2 3 10 16 60
1 2 5 6 14 60
1 3 4 5 13 60
1 2 4 8 15 64
1 2 3 11 17 66
1 2 5 7 15 70
1 2 4 9 16 72
1 3 4 6 14 72
1 2 4 10 17 80
1 2 5 8 16 80
1 2 6 7 16 84
1 3 4 7 15 84
1 2 5 9 17 90
1 3 5 6 15 90
1 2 6 8 17 96
1 3 4 8 16 96
1 3 5 7 16 105
1 3 4 9 17 108
1 3 5 8 17 120
1 4 5 6 16 120
2 3 4 5 14 120
1 3 6 7 17 126
1 4 5 7 17 140
2 3 4 6 15 144
2 3 4 7 16 168
2 3 5 6 16 180
2 3 4 8 17 192
2 3 5 7 17 210
2 4 5 6 17 240
           

From the above table, we can eliminate all rows with unique products. For example, if the women lived in room 105, Willard would have known the drive times without asking the further question. Eliminating combinations with unique products leaves us with the following table.

        sum product
1 2 3 8 14 48
1 2 4 6 13 48
1 2 3 10 16 60
1 2 5 6 14 60
1 3 4 5 13 60
1 2 4 9 16 72
1 3 4 6 14 72
1 2 4 10 17 80
1 2 5 8 16 80
1 2 6 7 16 84
1 3 4 7 15 84
1 2 5 9 17 90
1 3 5 6 15 90
1 2 6 8 17 96
1 3 4 8 16 96
1 3 5 8 17 120
1 4 5 6 16 120
2 3 4 5 14 120
           

When Willard asked if any of the women have to drive just one hour, he immediately knew the solution. Note that every group of products above contains more than one combination that has a ‘1’.Thus, if the answer were, ‘Yes’, he would not immediately know the solution. But, if the answer were ‘No’, there would be only one possible combination, the last in the table. We can conclude from the fact that Willard knew the combination immediately after hearing the answer, that the answer must have been 'No'.

Thus, the combination is {2, 3, 4, 5} and the women’s dorm room number is 120.

Incidentally, Pat Marino, in ResLife, tells me that there is no room #120 on campus. It's a good thing I didn't say that the students were at Hamilton College!