Wednesday, January 27, 2010

Domino Loop Any Time?

I don't know whether our box of Domino still has all the pieces or some of them have been lost. It is also possible that remainders of two different sets have been mixed together, as some pieces are yellowish and some white. On a complete Domino set, if you spread all the pieces randomly and then create a chain, by attaching pieces with the same number of dots together, without any planning, should it always end with the same number it started and allow you to close a chain?

Enter your answer on our Family Puzzle Marathon page. Solve three and get a prize!

Maura MacK said...

If the roads only run N/S and E/W, then she must drive a total of 49 miles this morning. H to B and back (10 miles) + H to F and back (4 miles) + H to C and back (20 miles) = H to W (15 miles) = 49.

However, if roads run along the hypotenuse of the triangles that can be superimposed on the diagram, then she would "only" have to drive 38.62 miles. She would follow this path: H to B (5 miles) + B to F (5.385 miles) + F to C (10.2 miles) + C to W (18.03) = 38.62.

Maria said...

Maura, this is fantastic solution, but you have entered it in the wrong puzzle. It belongs here. I can't give you a puzzle point, as Rachel T was the first to solve it correctly. But you can try the Domino puzzle above.

rachel_t said...

Hmmm... we only have princess dominos in our house and I know we aren't always able to close a loop or even make a line. I need to think of a mathematical explanation.

Rachel T

Anonymous said...

No. Depending on the maximum value, N, on the dominoes (and assuming that each domino only appears once in the set - so that there are (N choose 2) + N dominoes in the set), then you can end up with islands that cannot be connected. A simple counter-example for N=3 is:

11 12 23 31 which leaves the dominoes 33 and 22 left out (as islands)

rachel_t said...

Is it ok if there are pieces left over? If yes, I'm questioning my answer given Anonymous's example above. I know we have pieces left over, but since we usually make lines and not loops, I'm not certain if they would still close the loop.

Maria said...

Rachel was right, but unsure of her answer she forgot that to prove that some statement containing word "always" is wrong, you only need to demonstrate one counter-example.
For example:
"Is it true that all the children always come to school on time?"
"No, my son was late yesterday."

Anonymous creatively suggested a counter-example with 6 Domino pieces, with 3 dots maximum. If you randomly spread all 6 pieces and then assemble in a chain you may end up with: 11 12 23 31 which leaves the dominoes 33 and 22 left out.

So, one puzzle point for Anonymous...
And a new puzzle later today.

Maria said...

Wow, I just found this answer dated Wed, January 27th from Beth on the Facebook page:

no you will not always end with the same number. unless you have a full set. If pieces are missing like all the sixes then you will not be able to make a full chain .If only two pieces are missing you can make a chain but the ends will be different numbers.

one puzzle point for her as she was the very first one to answer.