The Riddler — How Many Bananas Does It Take To Lead A Camel To Market?

 

Took along hia­tus from the Rid­dler, but I’m back!

For those that aren’t famil­iar, The Rid­dler is a week­ly puz­zle series from Oliv­er Roed­er at FiveThir­tyEight that usu­al­ly involves some com­bi­na­tion of math, log­ic and prob­a­bil­i­ty.

This weeks FiveThir­tyEight Rid­dler, How Many Bananas Does it Take to Lead a Camel to Mar­ket?:

sYou have a camel and 3,000 bananas. (Because of course you do.) You would like to sell your bananas at the bazaar 1,000 miles away. Your loy­al camel can car­ry at most 1,000 bananas at a time. How­ev­er, it has an insa­tiable appetite and quite the nose for bananas — if you have bananas with you, it will demand one banana per mile trav­eled. In the absence of bananas on his back, it will hap­pi­ly walk as far as need­ed to get more bananas, loy­al beast that it is. What should you do to get the largest num­ber of bananas to the bazaar? What is that num­ber?

So just in order to get to the mar­ket, you need a min­i­mum of 1,000 bananas, and since that just gets you there with­out any bananas to sell, a more cre­ative solu­tion is required. Let’s say we split the 3,000 bananas into 3 piles — Pile A, Pile B, & Pile C. From there we move Pile A one mile, go back and do the same for Piles B & C.

Mile	Pile A	Pile B	Pile C
0	1000	1000	1000
1	999	999	999

This won’t get us any­where, since at mile 1,000 we’ll be left with 0 bananas in each. Let’s instead redis­trib­ute the Pile C bananas to Pile A and Pile B.

Mile	Pile A	Pile B	Pile C
0	1000	1000	1000
1	999	999	999
1	1000	1000	997

Now if we were to just move each pile for­ward with­out redis­tri­b­u­tion we’d end up with 2 bananas to sell at the mar­ket (1 from Pile A and 1 from Pile B). Let’s redis­trib­ute the bananas every mile, now it looks like:

Mile	Pile A	Pile B	Pile C
0	1000	1000	1000
1	999	999	999
1	1000	1000	997
2	999	999	996
2	1000	1000	995
3	999	999	994
3	1000	1000	992

A pat­tern starts to emerge, every mile takes 3 bananas, specif­i­cal­ly from Pile C. For now, let’s say:

Bananas in Pile C = 1000 — 3*Miles Trav­eled

Now let’s look at Mile 333, about when Pile C would be exhaust­ed.

Mile	Pile A	Pile B	Pile C
333	1000	1000	1
334	999	999	0

Since we only have 1 banana left in Pile C, we can’t redis­trib­ute it. So we’re now 667 miles away from our des­ti­na­tion, with 2 piles of 1,000 bananas each. Let’s con­tin­ue this redis­tri­b­u­tion strat­e­gy with Piles A and Pile B (redis­trib­ut­ing Pile B)

Mile	Pile A	Pile B	Pile C
334	999	999	0
334	1000	998	0
335	999	997	0
335	1000	995	0

Now anoth­er pat­tern emerges for Pile B. The amount of bananas in Pile B can be described as:

Bananas in Pile B = 1000 — 2*(Miles Trav­eled-333)

Using this we see that Pile B runs out bananas around mile 833.

Mile	Pile A	Pile B	Pile C
832	1000	2	0
833	999	1	0
833	1000	0	0

At this point, we can sim­ply just car­ry Pile A, feed­ing a banana to the camel every mile for the last 177 miles. The amount of bananas left when we reach the mar­ket is sim­ply 1000 — 177 = 833.

To answer the orig­i­nal Rid­dler ques­tion, the largest amount of bananas that we can get to the bazaar under the cur­rent cir­cum­stances is 833.