Place the numbers 1 to 100 in any order in a 10x10 array.

Create set A = the greatest number in each of the 10 rows.

Create set B = the least number in each of the 10 columns.

1. Demonstrate that the sets may or many not have numbers in common.

2. What is the smallest number that could be in both sets?

3. How many numbers could the two sets have in common?

Part 1) An example of no numbers in common: Put the numbers from 1-10 on one diagonal, and the numbers from 91-100 on the other. Then Set A = {91, 92, ..., 100) and Set B = {1,2,...10}

For an example of some numbers in common: See answer to part 2.

Part 2) The smallest number that can be in both sets is 10. One way to achieve this is to put the numbers from 1-10 on the first row. Then Set B = {1,2,...10} and Set A includes 10 from the first row, Clearly this is the minimum, as Set A can never include a number less than 10, because 9 can never be the minimum in a row of 10 numbers.

Part 3) The two sets can never have more than one number in common. To see this, assume that there is one number in common, and call it N. Then every number in N's column is greater than or equal to N, so no number less than N can be the maximum in its row, and therefore no number less than N can be in set A. Similarly, no number greater than N can be in set B. Therefore, N is the only number that the sets have in common.

*Edited on ***August 22, 2017, 8:32 pm**