# Who is sitting at the banker's left?

The Fall 1978 issue of Pi Mu Epsilon Journal included this problem, submitted by Pier Square.

Four men are playing bridge. Their names are Banker, Waiter, Baker, and Farmer, and, as it happens, each man’s name is another man’s job. Mr. Baker’s partner is the baker, Mr. Banker’s partner is the farmer, and the waiter sits at Mr. Farmer’s right. Who is sitting at the banker’s left?

(Note that in bridge, one's partner sits across from him.)

Problem was also published on futilitycloset.com, an excellent source of puzzles.

Mr. Baker's partner is the baker, not the waiter.
Mr. Banker's partner is the farmer, not the waiter.
Mr. Farmer's right is the waiter so Mr. Farmer's partner is not the waiter.
So Mr. Waiter's partner is the waiter.

Mr Farmer's right is the waiter so Mr Farmer's left is Mr. Waiter.
Mr. Banker's partner is the farmer so he is not Mr. Farmer.
Therefore, Mr. Banker's partner is Mr. Waiter (farmer).
So Mr. Waiter's partner (Mr. Banker) is the waiter.

Mr. Baker's partner must be Mr. Farmer (who is the baker).
Mr. Farmer is the baker; Mr. Banker is the waiter; Mr. Waiter is the farmer.
So the banker must be Mr. Baker.

So Mr. Baker's left is Mr. Farmer's right. He is Mr. Banker and he is the waiter.

• It's a nice solution, but I think the explanation is hampered by 2 things (1) the clash of names between Mr N and job n, I'd advise just aliasing the names as A, B, C, and D during the reasoning, so it reads easier and (2) the lack of partial diagram during the reasoning, as it requires juggling all the facts in one's head. I think ASCII diagrams would be fine (and cheap enough). Commented Aug 8 at 17:00

I like to start by numbering the assertions.

1. Each man's name is another man's job.
2. Mr. Baker’s partner is the baker.
3. Mr. Banker’s partner is the farmer.
4. The waiter sits at Mr. Farmer’s right.

I'll name the positions North, South, East, and West.

Without loss of generality, put Mr. Baker in the West position. By #2, that puts the baker in East. Let's try some names for the baker in East.

Let's start by assuming the baker is Mr. Banker. Then by #3, West (East's partner) is the farmer. With East and West being Mr. Banker and Mr. Baker, that leaves Mr. Farmer and Mr. Waiter as North and South, although not necessarily respectively. By #4, the waiter is to the right of Mr. Farmer, so the waiter is at either East or West. But we already know that the professions at East and West are the baker and the farmer. So our assumption that the baker is Mr. Banker led to a contradiction.

The next attempted assumption is that the baker (East) is Mr. Farmer. Then by #4, the waiter sits at North (East's right). By #1, North can't be both the Waiter and Mr. Waiter, and we've already placed Mr. Baker and Mr. Farmer, so the only remaining name for North is Mr. Banker. Then by #3, South is the farmer. We've placed three of the surnames, and then the only remaining name for South is Mr. Waiter.

So far, we've got:
North: Mr. Banker, the waiter
West: Mr. Baker, the ?
East: Mr. Farmer, the baker
South: Mr. Waiter, the farmer

The only remaining profession for West is the banker. And now we can answer the question of who is sitting to the banker's left. The banker is West, and to the left of West is North, and that's Mr. Banker, the waiter.