The following problem can be given as a FUN recreational problem to HS students or even younger: (I am sure that many of you already know it but my point is how to present it to HS students and perhaps even younger.)

Alice will say all but ONE of the elements of {1,...,10

^{10}}in some order.

Bob listens with the goal of figuring out the number. Bob cannot possibly store 10

^{10}numbers in his head. Help Bob out by giving him an algorithm which will not make his head explode.

This is an easy and fun puzzle. The answer is in the writeup I point to above.

The following variant is a bit harder but a bright HS student could get it: Same problem except that Alice leaves out TWO numbers.

The following variant is prob more appropriate for a HS math competition than for a FUN gathering of HS students: Same problem except that Alice leaves out THREE numbers.

The following variant may be easier because its harder: Alice leaves out k numbers, k a constant. Might be easier then the k=3 case since the solver knows to NOT use properties of 3.

I find it interesting that the k=1, k=2, and k≥ 3 cases are on different levels of hardness. I would like a more HS answer to the k≥ 3 case.

Here's a family of problems with different hardnesses for k=1,2,3,4,5: Given a planar graph, is it k-colorable? k=1: Is there an edge? k=2: Greedy algorithm. k=3: NP-hard. k=4: trivially yes; proof is somewhat long. k=5: proof is maybe two pages for a college student. (Is there an even easier proof that all planar graphs are 6-colorable?)

ReplyDeleteI think for k=6 there is an easier proof. It should follow from observation that every planar graph has a vertex of degree 5 (or less) via an easy induction.

DeleteHere's a family of problems with different hardnesses for k=1,2,3,4,5: Is a given planar graph k-colorable?

ReplyDeletek=1: Is there an edge?

k=2: Greedy algorithm

k=3: NP-hard

k=4: Trivially yes; proof is somewhat long and tedious.

k=5: Proof is maybe two pages for a college student.

k>=6: Even easier?

"Might be easier then the k=3 case since the solver knows to NOT use properties of 3."

ReplyDeleteWhere you write "then" I believe you mean "than". Also, you can avoid splitting the infinitive with "knows NOT to use" rather than "knows to NOT use".

Two remarks:

ReplyDelete1. One may argue that the case k=1 is the decisive one as the others are just technical embellishments using the same basic idea but more machinery (which you may not be familiar with).

2. A variant of the puzzle would be that Bob listens to a sequence of numbers and has to determine if they are all contiguous or contain holes. Same solution (remember the sum but just also remember min and max). What about naming all holes?

More variants:

ReplyDelete(1) Count the number of holes, or in the original setting the missing numbers from the interval [0,n]

(2) Allow duplication on the side of Alice, i.e. she may repeat some of the numbers.