We make cardboard boxes. We manufacture a variety of sizes and are concerned that storing them and transporting them will be a problem unless we can nest them into a reasonable number of nested stacks. The boxes are rectangular, but they are open at the top so their heights don't matter. If we call the smaller horizontal dimension of a box its width and the other dimension its length, a box can be nested in another box if its width is less than the other's width, and its length is less than the other's length. No diagonal nesting is allowed.

We have automated the manufacturing process so that random sized boxes are produced. We specify values a, p, and n and the machine produces n boxes with dimensions:

(a,a^2) (a^3,a^4) (a^5,a^6) ... (a^(2n-1),a^(2n)) where a^j denotes (a to the j power) mod p.

Create a class UnNestable that contains a method maxCount that is given the positive integral values a, p, and n and that returns the size of the largest unnestable collection of boxes, where a collection is unnestable if no two boxes from the collection can be nested.

The sequence of boxes can be generated by

int rv=1; for(int i=0; i<n; i++){ rv = (rv*a)%p; int x=rv; rv = (rv*a)%p; int y=rv; // process this box, // which has dimensions x and y }

Class: UnNestable Method: maxCount Parameters: int, int, int Returns: int Method signature: int maxCount(int a, int p, int n) (be sure your method is public)

The sequence may cycle, resulting in many boxes of the same dimensions. Duplicate boxes do count in the size of an unnestable collection

- p is between 2 and 2,000,000,000, inclusive.
- a is between 1 and p-1, inclusive.
- No power of a is divisible by p.
- a*p is less than or equal to 2,000,000,000
- n is between 1 and 10,000 inclusive

0) 10 15 3 Returns: 3 The random generator produces 10, (10*10)%15=10, 10*10%15=10, ... so the The three boxes are (10,10), (10,10), and (10,10). The entire collection is unnestable. 1) 10 17 4 Returns: 2 The sequence is 10, 100%17=15, 150%17=14, 140%17=4, 40%17=6, ... so the 4 boxes are (10, 15), (4, 14), (6, 9) and (5, 16). The non-nesting pairs are (10, 15) with (5, 16), (4, 14) with (6, 9), and (6, 9) with (5, 16). No collection of 3 of these boxes is pairwise unnestable. 2) 3 1000 3 Returns: 1 The boxes are (3,9), (27,81), and (243,729). The first one is nestable in both the others, and the second one is nestable in the third one. So there is no way to choose 2 of them which are unnestable. Of course, any collection of size one has the property that no two boxes from the collection can be nested. 3) 3 104729 10000 Returns: 9

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