More Refined Pricing

Joey is the founder and owner of Pure Diamonds & Pearls (PDP), a personal services provider referral agency. Joey has hired his cousin Tony as a management consultant. PDP has been charging all clients the same fixed price of $300. Tony advises Joey to vary his prices because different clients request a variety of different services and providers. Tony designs a new pricing formula P(m,n) > $0, prices always being positive, where the pricing factors m and n can each take on any integer value {…,-2, -1, 0, 1, 2,…} and the base price is P(0, 0) = $300. Tony’s formula is designed to obey the constraint P(m,n) = (P(m+1, n) + P(m-1, n) + P(m, n+1) + P(m, n-1))/4, so that each price is equal to the average of its nearest neighboring prices. Is there a lower bound on P(m,n) that is higher than $0? If so, can you determine the greatest lower bound? Is there any upper bound on P(m,n)? If so, can you determine the least upper bound?

Extra Credit: If instead of just two pricing factors there had been three or more pricing factors and an analogous local average constraint, what would your answers be to the previous questions?

Infinity Within Infinity Within Infinity?

This puzzle asks for a set of partitions of the positive integers such that:

• There are infinitely many partitions in the set.
• Any two of the partitions are disjoint.
• Each of the partitions contains infinitely many subsets.
• Every subset of every partition contains infinitely many numbers.

There are many ways to do this. Here is a construction submitted by Zack Murtha. Partition A_k depends on the kth prime number, p_k. The ith subset of A_k contains all the numbers divisible by (p_k)^(i-1) but no higher power of p_k.

So the first partition, A_1, has the following subsets:
{1,3,5,7,9,11,…} [Divisible by 2^0, but not 2^1]
{2,6,10,14,18,22,…} [Divisible by 2^1 but not 2^2]
{4,12,20,28,36,44,…} [Divisible by 2^2 but not 2^3]
{8,24,40,56,72,88,…} [Divisible by 2^3 but not 2^4]
etc.

The second partition, A_2, has the following subsets:
{1,2,4,5,7,8,…} [Divisible by 3^0 but not 3^1]
{3,6,12,15,21,24,…} [Divisible by 3^1 but not 3^2]
{9,18,36,45,63,72,…} [Divisible by 3^2 but not 3^3]
{27,54,108,…} [Divisible by 3^3 but not 3^4]
etc.

A_3 is the following:
{1,2,3,4,6,7,…}
{5,10,15,20,30,35,…}
{25,50,75,100,150,…}
{125,250,375,500,…}
etc.

The partitions will continue upward with the primes. Continuing in this manner will satisfy the requirements of the problem. Solutions were also submitted by Patrick Allen, Nathan Babcock, Roger Bovard, Daniel Chammas, Bob Conger, Ken Dailey, Blake Eastman, Akshar Gohil, Bruce Hackworth, Bill Hansen, Clive Keatinge, Ziru (James) Li, Robert Lee, David Lozinski, John McNulty, Zack Murtha, Jenni Prior, Brad Rosin, Dave Schofield, Cam Thomas, Geoff Tims, David Uhland and Yuchi Zhang.

AR Puzzle Editor Jon Evans is president of Convergent Actuarial Services, Inc. in Delray Beach, Florida.