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Domino arrangements (Posted on 2006-05-27) Difficulty: 5 of 5
Take the 15 smallest dominoes in a set (double blank through double four.)

In how many ways can they be arranged in a row such that the numbers on consecutive pieces match.

Count the two directions separately.

See The Solution Submitted by Jer    
Rating: 3.6667 (3 votes)

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Solution re: Pencil and paper solution | Comment 5 of 7 |
(In reply to Pencil and paper solution by Steve Herman)

You never multiply by 2 to get reversals, so the initial count should include both directions, which would lead to 24, rather than 12 chains that break into 5 and 5. Similarly the 48 that give a 4 and a 6 should be 96.

For the 3-row and 7-row, I actually get 144.  This would be 72 if you didn't count reflections, but I see you record this as 36. Listed, these are:

40|01|13|30|02|21|14|42|23|34
40|01|13|30|02|21|14|43|32|24
40|01|13|30|02|23|34|41|12|24
40|01|13|30|02|23|34|42|21|14
40|01|13|30|02|24|41|12|23|34
40|01|13|30|02|24|43|32|21|14
20|01|13|30|04|41|12|23|34|42
20|01|13|30|04|41|12|24|43|32
20|01|13|30|04|42|21|14|43|32
20|01|13|30|04|42|23|34|41|12
20|01|13|30|04|43|32|21|14|42
20|01|13|30|04|43|32|24|41|12
30|01|14|40|02|21|13|32|24|43
30|01|14|40|02|21|13|34|42|23
30|01|14|40|02|23|31|12|24|43
30|01|14|40|02|23|34|42|21|13
30|01|14|40|02|24|43|31|12|23
30|01|14|40|02|24|43|32|21|13
20|01|14|40|03|31|12|23|34|42
20|01|14|40|03|31|12|24|43|32
20|01|14|40|03|32|21|13|34|42
20|01|14|40|03|32|24|43|31|12
20|01|14|40|03|34|42|21|13|32
20|01|14|40|03|34|42|23|31|12
40|02|21|10|03|31|14|42|23|34
40|02|21|10|03|31|14|43|32|24
40|02|21|10|03|32|24|41|13|34
40|02|21|10|03|32|24|43|31|14
40|02|21|10|03|34|41|13|32|24
40|02|21|10|03|34|42|23|31|14
30|02|21|10|04|41|13|32|24|43
30|02|21|10|04|41|13|34|42|23
30|02|21|10|04|42|23|31|14|43
30|02|21|10|04|42|23|34|41|13
30|02|21|10|04|43|31|14|42|23
30|02|21|10|04|43|32|24|41|13
40|02|23|30|01|12|24|41|13|34
40|02|23|30|01|12|24|43|31|14
40|02|23|30|01|13|34|41|12|24
40|02|23|30|01|13|34|42|21|14
40|02|23|30|01|14|42|21|13|34
40|02|23|30|01|14|43|31|12|24
10|02|23|30|04|41|12|24|43|31
10|02|23|30|04|41|13|34|42|21
10|02|23|30|04|42|21|13|34|41
10|02|23|30|04|42|21|14|43|31
10|02|23|30|04|43|31|12|24|41
10|02|23|30|04|43|31|14|42|21
30|02|24|40|01|12|23|31|14|43
30|02|24|40|01|12|23|34|41|13
30|02|24|40|01|13|32|21|14|43
30|02|24|40|01|13|34|41|12|23
30|02|24|40|01|14|43|31|12|23
30|02|24|40|01|14|43|32|21|13
10|02|24|40|03|31|12|23|34|41
10|02|24|40|03|31|14|43|32|21
10|02|24|40|03|32|21|13|34|41
10|02|24|40|03|32|21|14|43|31
10|02|24|40|03|34|41|12|23|31
10|02|24|40|03|34|41|13|32|21
40|03|31|10|02|21|14|42|23|34
40|03|31|10|02|21|14|43|32|24
40|03|31|10|02|23|34|41|12|24
40|03|31|10|02|23|34|42|21|14
40|03|31|10|02|24|41|12|23|34
40|03|31|10|02|24|43|32|21|14
20|03|31|10|04|41|12|23|34|42
20|03|31|10|04|41|12|24|43|32
20|03|31|10|04|42|21|14|43|32
20|03|31|10|04|42|23|34|41|12
20|03|31|10|04|43|32|21|14|42
20|03|31|10|04|43|32|24|41|12
40|03|32|20|01|12|24|41|13|34
40|03|32|20|01|12|24|43|31|14
40|03|32|20|01|13|34|41|12|24
40|03|32|20|01|13|34|42|21|14
40|03|32|20|01|14|42|21|13|34
40|03|32|20|01|14|43|31|12|24
10|03|32|20|04|41|12|24|43|31
10|03|32|20|04|41|13|34|42|21
10|03|32|20|04|42|21|13|34|41
10|03|32|20|04|42|21|14|43|31
10|03|32|20|04|43|31|12|24|41
10|03|32|20|04|43|31|14|42|21
20|03|34|40|01|12|23|31|14|42
20|03|34|40|01|12|24|41|13|32
20|03|34|40|01|13|32|21|14|42
20|03|34|40|01|13|32|24|41|12
20|03|34|40|01|14|42|21|13|32
20|03|34|40|01|14|42|23|31|12
10|03|34|40|02|21|13|32|24|41
10|03|34|40|02|21|14|42|23|31
10|03|34|40|02|23|31|12|24|41
10|03|34|40|02|23|31|14|42|21
10|03|34|40|02|24|41|12|23|31
10|03|34|40|02|24|41|13|32|21
30|04|41|10|02|21|13|32|24|43
30|04|41|10|02|21|13|34|42|23
30|04|41|10|02|23|31|12|24|43
30|04|41|10|02|23|34|42|21|13
30|04|41|10|02|24|43|31|12|23
30|04|41|10|02|24|43|32|21|13
20|04|41|10|03|31|12|23|34|42
20|04|41|10|03|31|12|24|43|32
20|04|41|10|03|32|21|13|34|42
20|04|41|10|03|32|24|43|31|12
20|04|41|10|03|34|42|21|13|32
20|04|41|10|03|34|42|23|31|12
30|04|42|20|01|12|23|31|14|43
30|04|42|20|01|12|23|34|41|13
30|04|42|20|01|13|32|21|14|43
30|04|42|20|01|13|34|41|12|23
30|04|42|20|01|14|43|31|12|23
30|04|42|20|01|14|43|32|21|13
10|04|42|20|03|31|12|23|34|41
10|04|42|20|03|31|14|43|32|21
10|04|42|20|03|32|21|13|34|41
10|04|42|20|03|32|21|14|43|31
10|04|42|20|03|34|41|12|23|31
10|04|42|20|03|34|41|13|32|21
20|04|43|30|01|12|23|31|14|42
20|04|43|30|01|12|24|41|13|32
20|04|43|30|01|13|32|21|14|42
20|04|43|30|01|13|32|24|41|12
20|04|43|30|01|14|42|21|13|32
20|04|43|30|01|14|42|23|31|12
10|04|43|30|02|21|13|32|24|41
10|04|43|30|02|21|14|42|23|31
10|04|43|30|02|23|31|12|24|41
10|04|43|30|02|23|31|14|42|21
10|04|43|30|02|24|41|12|23|31
10|04|43|30|02|24|41|13|32|21
40|01|12|20|03|31|14|42|23|34
40|01|12|20|03|31|14|43|32|24
40|01|12|20|03|32|24|41|13|34
40|01|12|20|03|32|24|43|31|14
40|01|12|20|03|34|41|13|32|24
40|01|12|20|03|34|42|23|31|14
30|01|12|20|04|41|13|32|24|43
30|01|12|20|04|41|13|34|42|23
30|01|12|20|04|42|23|31|14|43
30|01|12|20|04|42|23|34|41|13
30|01|12|20|04|43|31|14|42|23
30|01|12|20|04|43|32|24|41|13


I think your factor of 4 comes from not only discounting the complete reversals of, for example, the two below, but also from not counting even the two below as separate, where only the 3-segment is reversed :

40|01|13|30|02|21|14|42|23|34
40|03|31|10|02|21|14|42|23|34

their complete reversals being
20|03|31|10|04|43|32|24|41|12
20|01|13|30|04|43|32|24|41|12

Counting these four lines as one, and all similar situations results in the factor of 4 undercount.

(Yes, I used a computer to get the above list.)

Multiplying this (24+96+144=264) by 32*15 we get 126,720, which I believe is the correct answer.

Edited on May 28, 2006, 2:16 pm
  Posted by Charlie on 2006-05-28 11:45:15

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