# Sequence of domino tiles - next term's difference of pips

The problem is as follows:

A set of domino tiles is placed with one next to another as indicated in the diagram below. If the upper half of the next piece is labeled x and the lower half is labeled as y, then find x-y. The choices given in my book are follows:

1. 2
2. -3
3. 5
4. -5

I found this riddle in my Reason and Logic book from the 2000s. From the style I believe it is adapted from a reprinted version of Martin Gardner's 50's book on Recreational Puzzles

I remember similar puzzles which asked for the least amount of domino pieces which need to be flipped vertically to make the count of pips on the upper and lower halves equal. My method of solving was to sum all the dots, divide this number by two, and see which tiles can be rotated in order to achieve equality.

But in this case I need to find some sort of logic to the ordering of the tiles (or at least I think so), and they appear to be in a random order, which confuses me.

I believe that this question (although it doesn't explicitly say) intends to imply that the set is a double-six. Hence it will have 28 tiles and 126 pieces.

So far I noticed that the third tile starting from the left is swapped upside down in the fourth place, right next to the double zero tile. This could mean that this domino comes from either the set of 3 or the set of 4. But again I couldn't spot anything other than that.

Can someone explain what kind of logic should be used to find the next term in this sequence? Is it just trial and error? I tried finding the difference between the pips in the upper and lower halves, but this didn't help much. Can someone please guide me to a solution? What strategy should be used?

• Three further aspects confuse me: 1) What are "126 pieces"? Total number of pips, for instance, seems to be 168 in a double-six set. 2) A double-six set doesn't have two [3|4]s. 3) The two [3|4]s are rotated mirror images. And i so far see only two consistent patterns: 1) Neither top nor bottom row has duplicates. 2) Total pips on each domino here has 0 remainder when divided by 7.
– humn
Nov 1 '20 at 20:28
• My answer would be exactly the same with the same ratinalization. It's not very original but simple and fits rather well. Apr 9 at 15:23

Option 4 (-5)

Why?

We can see that the double zero domino flips the domino before it upside-down. Therefore, the next domino must be the 2nd domino(the domino before the double zero domino) flipped upside-down. Hence, x=1 and y=6, and so x - y = 1 - 6 = -5.

• That seems like a bit of a stretch! Nov 2 '20 at 3:29

Without any further context, the question is on par with number sequence "guessing game", and you won't find any logical path or strategy that leads to the solution. I'm specifically pointing this out because the OP asked for logic and strategy.

But, to my intuition the intended answer looks like

-5 (option 4), with x=1,y=6

which is the same answer as mccraft's, but for a different reason:

Look at the dominoes at the odd positions. From left to right, the top number is decreasing (5 - 4 - 3) while the bottom is increasing (2 - 3 - 4). Now look at the dominoes at the even positions. We can see the bottom as decreasing (1 - 0). For the top, the 6 is supposed to increase, but there's no 7 pips, so it is natural to assume it wraps back to 0 instead. Therefore, I think the top increases to 1, and the bottom "decreases" (wrapping) to 6.

• I feared you would say this. But in end its what it is. Probably the best recommendation which I was intending to ask was. Before getting into panic, What should someone start to look on?. I gave some preface in my question regarding on what do I recall on domino tiles and the number of pips, which I thought it could help, but it seems that in this case it didn't. By the way I did read the article you mentioned but since the peculiar nature of this kind of puzzle I doubt the OEIS would had this sequence, although I must admit that your observation was very neat. Nov 8 '20 at 7:44
• Out of curiosity, did it took you too long to notice the pattern in those pieces?. I didn't really figured out that what the author might had intended was to keep the increasing the pips in the upper portion of the domino and decreasing the number in the lower portion. Should it be this as a good rule of thumb?. Or is it just a guessing game?. In this case you didn't flipped the tiles, does this means that a good strategy would be leave untouched the pieces and focus in the number of pips first?. Nov 8 '20 at 7:47
• @ChrisSteinbeckBell For a pattern-spotting puzzle, some recurring themes include increasing and decreasing numbers with equal gaps, and separate (but possibly related) patterns for even and odd places (1st, 3rd, 5th, ... and 2nd, 4th, 6th... as two groups). When each piece has two or more parts (like upper and lower cells of a domino), considering the two separately also helps. Wrapping around while counting could be easier to spot if you have some experience with basic number theory (modular arithmetic). Nov 9 '20 at 0:36
• I don't know if any specific strategy will help tackle this kind of puzzles. Sometimes it has to do with the individual values, in the others it has to do with the whole piece (or even the whole table of givens). I'd suggest to get more familiar with various kinds of patterns, and try to see the trees as well as the forest and find where a pattern emerges. Nov 9 '20 at 0:41

I think that these are the more correct answers:

0, $$x=0, y=0$$ or 9, $$x=8,y=-1$$

Because:

On the top row of the dominoes, the value is added and subtracted with consecutive numbers, which means after an addition, there is subtraction, and addition comes first. However, the displayed value cannot be greater than 6 or smaller than 0^. For numbers greater than 6 or smaller than 0, the displayed number is represented with 0^. The first displayed number is its actual value which is 5, also for the second which is 6, and $$5+1=6$$. The third displayed number is its actual value which is 4, and $$6-2=4$$. The forth displayed number is 0, but the actual value is 7, because $$4+3=7$$. The fifth displayed number is its actual value which is 3, and $$7-4=3$$. For the next number(which is $$x$$), the actual value is $$3+5=8$$. However, the displayed value cannot be greater than 6, so $$x=0$$. For the bottom row, the rules are the same, just that subtraction comes first. Using the rules, we know that the actual value of $$y=-1$$, however because of the rules, the displayed value of $$y=0$$. Therefore, by subtracting the two displayed values, the answer is 0. However, the question did not mention that $$x$$ and $$y$$ are in their displayed values or their actual values. We can also see that in the question, there are cases in which the question subtracted or added a number's actual value(like as in previously mentioned, $$7-4=3$$). So, if we subtract the actual values of $$x$$ and $$y$$, we get 9.