If you’ve just joined us, our last episode ended while we were still sitting in a restaurant, pondering our situation: We had NINE (9) choices for entrees, from which we could pick ANY 2 for the daily special. The menu suggested there were 81 combinations from which to choose, and we weren’t sure we agreed.
Responses to the set-up in that last column were both fun and varied. And, by the way, noticeably free from the oh-god-I’m-supposed-to-know-the-answer feeling often encountered in a classroom. Part of the point, naturally.
When the dust settled, the majority (but not all) of respondents thought the restaurant was wrong. I received proposed solutions of 36, 45, and 48, as well as the 81. Surprisingly, I did NOT receive a submission of 72, which I had expected. See below.
One reader submitted her answer of 36. Here is her (slightly edited) approach:
I used an easier problem of 4 foods so I could find the pattern. I used ABCD and saw that there were 3 choices for A, 2 for choice B, 1 for C, and no new choices for D. So, with 9 choices it would be 8+7+6 etc, or 36 different choices.
Did you get the idea? This is a nice approach, and I particularly liked the ‘used an easier problem’ as a first step!
Another reader also arrived at 36, and further noted that if you allow two of the same choice (chicken and chicken – remember?), that would add nine meals to the total, bringing the total choices to 45.
A reader who is also a teacher volunteered that “problems like this made good homework problems and got family members involved, as well!”
So. Where did the ‘81’ answer come from, and what’s ‘wrong’ with it? Maybe nothing! It appears the menu designer multiplied 9 means times 9 (each of 9 entrees can be paired with each of the other nine, allowing repeats?) to get 81.
This is what often happens with solvers. They usually decide not to allow repeats, then pair each entree with 8 others, giving the 72 (9×8) meals I had anticipated I might get. Note that, for my money, this is good reasoning!! Where’s the hidden flaw?
When the above happens, the solver often proudly displays 9 columns of 8 meals each. One can then often point to a choice (say, AC in column 1) and then find its counterpart elsewhere (CA in column 3) and ask, “is it a different choice if you turn the plate around?”
The response is usually a broad grin (and a forehead slap?) which indicates the light coming on! At this point, they often add, “OK, divide 72 in half and you get 36!” Another very good approach! (One submitter above even noted “I decided chicken and grits is the same as grits and chicken.”)
Interestingly, every so often, the (smart-aleck?) response to that plate-turning question is “Yes, it is different. I eat my meals from left to right!” And with that condition, 72 (or 81) becomes a ‘right’ answer! Ah, conditions!
So, in real life, with these variables, we usually agree on 36 or 45 possibilities.
I must add this fun PS: A teacher once sent me an ad from the St. Louis paper featuring a similar display for the same restaurant chain. The ad proclaimed: “Here are 3 of the 54 choices possible!” 54?! WHERE did the 54 came from?! Any thoughts out there?