# Five professors and nine dishes

Here's yet another puzzle adapted from a puzzle book (in my case, with edits to make some of the specifications of the puzzle more clear because I didn't really understand them the first time I read it).

Five professors are spending three days on a retreat, and every day they eat dinner at the same restaurant. This restaurant has a special "roulette menu" with nine items, in which you choose a menu item from 1 to 9 that corresponds to one of the nine available items, but you don't know which number corresponds to which item. When the items are ordered, they are brought to the table in no particular order, so if a group orders multiple items at a time, they cannot figure out individually which item was ordered by whom.

These professors have tasked themselves with figuring out which number corresponds to which item in three days regardless. Each professor will order one item per day. How should they plan their ordering of the items in order to find out which number corresponds to which item for all items?

• To make the puzzle work as intended, I think it's also necessary to specify that the orders at each meal are "scrambled" so that no individual dish is associated with a single professor. Otherwise they could identify five dishes per meal: e.g. the first professor orders dish #1 and knows that whatever dish he gets is #1, and so on. – David Z Dec 25 '14 at 15:15
• Real-world application: Chinese restaurants – Lopsy Dec 25 '14 at 23:50
• @Lopsy: Assuming you don't know Chinese, anyway. – Joe Z. Dec 26 '14 at 3:37
• @DavidZ That's what I meant with the "no particular order" bit. – Joe Z. Dec 26 '14 at 3:38
• @JoeZ. ah, well I didn't get that from your wording. It's possible - in fact it is exactly how most restaurants work - to bring the dishes to the table in arbitrary order (i.e. it's arbitrary which one comes first, which one comes second) and still have each one placed in front of the professor who ordered it. – David Z Dec 26 '14 at 19:14

Okay, 15 orders, 9 dishes, that means 6 dupes. So 3 dishes will only be ordered once, obviously never on the same day. Now to split the six dupes so they can all be detected.

The first 3:

Day 1: 1   3
Day 2: 1 2
Day 3:   2 3


The dish that shows on day 1 and 2 is #1, and so on.

Day 1: 1   3 4 4
Day 2: 1 2   5 5
Day 3:   2 3 6 6


So the dish that's doubled on day 1 is #4, etc.

And now the last three:

Day 1: 1 7 3 4 4
Day 2: 1 2 8 5 5
Day 3: 9 2 3 6 6


The dish that only shows up on day 1 is #7, and so on.

And then, secure in their triumph, they can leave five bad Yelp reviews warning people away from this crazy place.

• Very nicely explained. On the contrary to the Yelp reviews, though: now they can publish and sell a guidebook! – Josh Caswell Dec 25 '14 at 3:59
• Gah beat me by seconds! :) – McMagister Dec 25 '14 at 4:00
• Thanks very much, Josh. And better luck next time, McMagister.:) – Len Pitre Dec 25 '14 at 6:11
• They could also identify a 10th item that nobody orders. That is assuming the set of dishes is known but not their numbers. – Florian F Dec 26 '14 at 13:25
• @McMagister - For once somebody beat you to it! :-) – Rand al'Thor Dec 28 '14 at 19:30