GeekDad Puzzle of the Week Solution – Backsplash Backlash

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Yes, this was really our kitchen last weekend.

Here is this past week’s puzzle as originally posted:

As we enter week two of our kitchen renovation project, my wife Allison and I have decided that some decisions are joint decisions (we had to agree on which model fridge to purchase) and some are single-spouse decisions (I got to decide how to rewire sockets, Allison selected the cabinets.)

The challenges come when one partner thinks a joint decision is single-spouse, or vice-versa. When laying out a grid for the backsplash we selected, I thought it was reasonable to measure and mark the grid down, then across; Allison suggested I measure and mark across, and then down. “Good thing that there are only two ways to do this,” I said, and then started thinking…

If the backsplash were simply a 3″ x 2″ grid, there would be some ten ways to mark off each horizontal and vertical measurement:

However, our backsplash is a little larger — it’s 12″ x 120″. This week’s assignment, if you choose to accept it, is to determine the number of distinct ways we can mark off or traverse the 12″ x 120″ grid, as shown above. So that calculating technology is not a deciding factor, feel free to round to the nearest quadrillion.

One way to think about the backsplash traversal is as the set of moves it requires. In the 3 x 2 grid, above, completely independent of the order in which they were taken, we needed to move three times from left to right, and two times from top to bottom. A total of 5 moves (3 + 2) were required. If we look only at the horizontal moves (and assume the rest were vertical), the solution for this grid is simply the number of ways from which you could choose 3 items from a set of 5, or C(5,3). If we were to look only at the two vertical moves, the answer would be C(5,2) which is the same value — the 10 we saw above.

The numbers do get a little more interesting at the scale of the full backsplash, but your favorite spreadsheet program or online computational knowledge engine should be able to get you the answer straight away. Of the 120 + 12 = 132 moves necessary to grid out our backsplash, 12 were vertical. The value of 132 choose 12 or C(132, 12) is 34,898,565,177,533,200 or thirty four quadrillion, eight hundred ninety eight trillion, five hundred sixty five billion, one hundred seventy seven million, five hundred thirty three thousand, two hundred different ways to grid out our backsplash.

Congratulations to Michael Gaisser for entering a correct submission. His entry was drawn at random from all of the reasonably correct responses, and this week’s $50 ThinkGeek Gift Certificate will be coming his way!

Thanks to everyone that submitted a response to this week’s puzzle.

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