perplexus.info :: Logic : Two Logicians Lockout

Two Logicians Lockout (Posted on 2007-03-13)

Two different integers from 1 to N are chosen. Sam is given the sum and Pat is given the product. Sam and Pat take turns stating whether they can determine the numbers

Sam: I don't know the numbers.
Pat: I don't know the numbers.
Sam: I don't know the numbers.
Pat: I don't know the numbers.
.
.
.
.
Sam and Pat continue like this until one of them realizes that it is impossible for either of them to determine the numbers. What is the smallest possible N for which this can happen?

See The Solution Submitted by Brian Smith

Rating: 4.0000 (1 votes)

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re: computer solution -- more

| Comment 3 of 10 |

(In reply to computer solution by Charlie)

By the way, the ambiguous cases are listed after the set of possible conclusions by Sam and Pat. When N exceeds 9, the impossibility conclusion can come earlier than turn 10 (5 for each), as shown in the following. I've repeated N=9 and continued up to N=12.

9 Sam says 1 1 2 1 in turn 1
9 Sam says 1 2 3 2 in turn 1
9 Sam says 8 9 17 72 in turn 1
9 Sam says 9 9 18 81 in turn 1
9 Pat says 1 3 4 3 in turn 2
9 Pat says 1 5 6 5 in turn 2
9 Pat says 1 7 8 7 in turn 2
9 Pat says 2 5 7 10 in turn 2
9 Pat says 2 7 9 14 in turn 2
9 Pat says 3 5 8 15 in turn 2
9 Pat says 3 7 10 21 in turn 2
9 Pat says 3 9 12 27 in turn 2
9 Pat says 4 5 9 20 in turn 2
9 Pat says 4 7 11 28 in turn 2
9 Pat says 4 8 12 32 in turn 2
9 Pat says 5 5 10 25 in turn 2
9 Pat says 5 6 11 30 in turn 2
9 Pat says 5 7 12 35 in turn 2
9 Pat says 5 8 13 40 in turn 2
9 Pat says 5 9 14 45 in turn 2
9 Pat says 6 7 13 42 in turn 2
9 Pat says 6 8 14 48 in turn 2
9 Pat says 6 9 15 54 in turn 2
9 Pat says 7 7 14 49 in turn 2
9 Pat says 7 8 15 56 in turn 2
9 Pat says 7 9 16 63 in turn 2
9 Pat says 8 8 16 64 in turn 2
9 Sam says 2 2 4 4 in turn 3
9 Sam says 4 9 13 36 in turn 3
9 Sam says 6 6 12 36 in turn 3
9 Pat says 1 4 5 4 in turn 4
9 Sam says 2 3 5 6 in turn 5
9 Pat says 1 6 7 6 in turn 6
9 Sam says 3 4 7 12 in turn 7
9 Pat says 2 6 8 12 in turn 8
9 Sam says 4 4 8 16 in turn 9
9 Pat says 2 8 10 16 in turn 10
9
1 8 9 8
1 9 10 9
2 4 6 8
2 9 11 18
3 3 6 9
3 6 9 18
3 8 11 24
4 6 10 24

10 Sam says 1 1 2 1 in turn 1
10 Sam says 1 2 3 2 in turn 1
10 Sam says 9 10 19 90 in turn 1
10 Sam says 10 10 20 100 in turn 1
10 Pat says 1 3 4 3 in turn 2
10 Pat says 1 5 6 5 in turn 2
10 Pat says 1 7 8 7 in turn 2
10 Pat says 2 7 9 14 in turn 2
10 Pat says 3 5 8 15 in turn 2
10 Pat says 3 7 10 21 in turn 2
10 Pat says 3 9 12 27 in turn 2
10 Pat says 4 7 11 28 in turn 2
10 Pat says 4 8 12 32 in turn 2
10 Pat says 5 5 10 25 in turn 2
10 Pat says 5 7 12 35 in turn 2
10 Pat says 5 9 14 45 in turn 2
10 Pat says 5 10 15 50 in turn 2
10 Pat says 6 7 13 42 in turn 2
10 Pat says 6 8 14 48 in turn 2
10 Pat says 6 9 15 54 in turn 2
10 Pat says 6 10 16 60 in turn 2
10 Pat says 7 7 14 49 in turn 2
10 Pat says 7 8 15 56 in turn 2
10 Pat says 7 9 16 63 in turn 2
10 Pat says 7 10 17 70 in turn 2
10 Pat says 8 8 16 64 in turn 2
10 Pat says 8 9 17 72 in turn 2
10 Pat says 8 10 18 80 in turn 2
10 Pat says 9 9 18 81 in turn 2
10 Sam says 2 2 4 4 in turn 3
10 Sam says 4 10 14 40 in turn 3
10 Pat says 1 4 5 4 in turn 4
10 Pat says 5 8 13 40 in turn 4
10 Sam says 2 3 5 6 in turn 5
10 Pat says 1 6 7 6 in turn 6
10
1 8 9 8
1 9 10 9
1 10 11 10
2 4 6 8
2 5 7 10
2 6 8 12
2 8 10 16
2 9 11 18
2 10 12 20
3 3 6 9
3 4 7 12
3 6 9 18
3 8 11 24
3 10 13 30
4 4 8 16
4 5 9 20
4 6 10 24
4 9 13 36
5 6 11 30
6 6 12 36

11 Sam says 1 1 2 1 in turn 1
11 Sam says 1 2 3 2 in turn 1
11 Sam says 10 11 21 110 in turn 1
11 Sam says 11 11 22 121 in turn 1
11 Pat says 1 3 4 3 in turn 2
11 Pat says 1 5 6 5 in turn 2
11 Pat says 1 7 8 7 in turn 2
11 Pat says 1 11 12 11 in turn 2
11 Pat says 2 7 9 14 in turn 2
11 Pat says 2 11 13 22 in turn 2
11 Pat says 3 5 8 15 in turn 2
11 Pat says 3 7 10 21 in turn 2
11 Pat says 3 9 12 27 in turn 2
11 Pat says 3 11 14 33 in turn 2
11 Pat says 4 7 11 28 in turn 2
11 Pat says 4 8 12 32 in turn 2
11 Pat says 4 11 15 44 in turn 2
11 Pat says 5 5 10 25 in turn 2
11 Pat says 5 7 12 35 in turn 2
11 Pat says 5 9 14 45 in turn 2
11 Pat says 5 10 15 50 in turn 2
11 Pat says 5 11 16 55 in turn 2
11 Pat says 6 7 13 42 in turn 2
11 Pat says 6 8 14 48 in turn 2
11 Pat says 6 9 15 54 in turn 2
11 Pat says 6 10 16 60 in turn 2
11 Pat says 6 11 17 66 in turn 2
11 Pat says 7 7 14 49 in turn 2
11 Pat says 7 8 15 56 in turn 2
11 Pat says 7 9 16 63 in turn 2
11 Pat says 7 10 17 70 in turn 2
11 Pat says 7 11 18 77 in turn 2
11 Pat says 8 8 16 64 in turn 2
11 Pat says 8 9 17 72 in turn 2
11 Pat says 8 10 18 80 in turn 2
11 Pat says 8 11 19 88 in turn 2
11 Pat says 9 9 18 81 in turn 2
11 Pat says 9 10 19 90 in turn 2
11 Pat says 9 11 20 99 in turn 2
11 Pat says 10 10 20 100 in turn 2
11 Sam says 2 2 4 4 in turn 3
11 Sam says 4 10 14 40 in turn 3
11 Pat says 1 4 5 4 in turn 4
11 Pat says 5 8 13 40 in turn 4
11 Sam says 2 3 5 6 in turn 5
11 Pat says 1 6 7 6 in turn 6
11
1 8 9 8
1 9 10 9
1 10 11 10
2 4 6 8
2 5 7 10
2 6 8 12
2 8 10 16
2 9 11 18
2 10 12 20
3 3 6 9
3 4 7 12
3 6 9 18
3 8 11 24
3 10 13 30
4 4 8 16
4 5 9 20
4 6 10 24
4 9 13 36
5 6 11 30
6 6 12 36

12 Sam says 1 1 2 1 in turn 1
12 Sam says 1 2 3 2 in turn 1
12 Sam says 11 12 23 132 in turn 1
12 Sam says 12 12 24 144 in turn 1
12 Pat says 1 3 4 3 in turn 2
12 Pat says 1 5 6 5 in turn 2
12 Pat says 1 7 8 7 in turn 2
12 Pat says 1 11 12 11 in turn 2
12 Pat says 2 7 9 14 in turn 2
12 Pat says 2 11 13 22 in turn 2
12 Pat says 3 5 8 15 in turn 2
12 Pat says 3 7 10 21 in turn 2
12 Pat says 3 9 12 27 in turn 2
12 Pat says 3 11 14 33 in turn 2
12 Pat says 4 7 11 28 in turn 2
12 Pat says 4 8 12 32 in turn 2
12 Pat says 4 11 15 44 in turn 2
12 Pat says 5 5 10 25 in turn 2
12 Pat says 5 7 12 35 in turn 2
12 Pat says 5 9 14 45 in turn 2
12 Pat says 5 10 15 50 in turn 2
12 Pat says 5 11 16 55 in turn 2
12 Pat says 6 7 13 42 in turn 2
12 Pat says 6 9 15 54 in turn 2
12 Pat says 6 11 17 66 in turn 2
12 Pat says 7 7 14 49 in turn 2
12 Pat says 7 8 15 56 in turn 2
12 Pat says 7 9 16 63 in turn 2
12 Pat says 7 10 17 70 in turn 2
12 Pat says 7 11 18 77 in turn 2
12 Pat says 7 12 19 84 in turn 2
12 Pat says 8 8 16 64 in turn 2
12 Pat says 8 10 18 80 in turn 2
12 Pat says 8 11 19 88 in turn 2
12 Pat says 8 12 20 96 in turn 2
12 Pat says 9 9 18 81 in turn 2
12 Pat says 9 10 19 90 in turn 2
12 Pat says 9 11 20 99 in turn 2
12 Pat says 9 12 21 108 in turn 2
12 Pat says 10 10 20 100 in turn 2
12 Pat says 10 11 21 110 in turn 2
12 Pat says 10 12 22 120 in turn 2
12 Pat says 11 11 22 121 in turn 2
12 Sam says 2 2 4 4 in turn 3
12 Sam says 3 12 15 36 in turn 3
12 Sam says 6 12 18 72 in turn 3
12 Pat says 1 4 5 4 in turn 4
12 Pat says 8 9 17 72 in turn 4
12 Sam says 2 3 5 6 in turn 5
12 Sam says 5 12 17 60 in turn 5
12 Pat says 1 6 7 6 in turn 6
12 Pat says 6 10 16 60 in turn 6
12 Sam says 4 12 16 48 in turn 7
12 Pat says 6 8 14 48 in turn 8
12
1 8 9 8
1 9 10 9
1 10 11 10
1 12 13 12
2 4 6 8
2 5 7 10
2 6 8 12
2 8 10 16
2 9 11 18
2 10 12 20
2 12 14 24
3 3 6 9
3 4 7 12
3 6 9 18
3 8 11 24
3 10 13 30
4 4 8 16
4 5 9 20
4 6 10 24
4 9 13 36
4 10 14 40
5 6 11 30
5 8 13 40
6 6 12 36

Posted by Charlie on 2007-03-14 09:13:38

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