The Hamilton Puzzler The Continental Conundrum The Befuddled Blue
Logic Puzzle #2: Solutions Page
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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!
Puzzle #2 (November 2008)
Thanksgiving is coming. Quadmates Judith, Penelope, Ruth, and Virginia are all driving home. Each student has her own car. Willard has come to their dorm room to say goodbye.
Willard asks, “How long do you each have to drive to get home?”
The women reply as follows:
We each have a different whole number of hours to drive.
The sum of these four numbers is less than eighteen.
The product of these numbers is our dorm room number.
Judith has the shortest drive, followed by Penelope, Ruth and Virginia, in that order.
Willard thinks for a little while, and scribbles some notes on a pad, but can not determine the four numbers.
He asks whether any one of the women has to drive just one hour to get home.
The women answer him, and he immediately knows the four numbers.
Questions:
1. How long does each woman have to drive to get home?
2. What is the women’s dorm room number?
Solutions must include an explanation.
The Puzzler's version of the winning solution:
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!
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