06 May '05 10:45

The set (1,2,3,4) can be partitioned into two subsets (1,4) and (2,3) of the same size. Note that 1 + 4 = 2 + 3

Qa. Find the next whole number n, above 4, for which the set (1,2,...,n) can be partitioned into two subsets S and T of the same size, with the sum of the numbers in S equal to the sum of the numbers in T. (this is farily simple as n can't be odd, and the sum of 1,...n can't be odd either)

Qb. Find all partitions in a with the ADDITIONAL property that the sum of the squares of the numbers in S equals the sum of the squares of the numbers in T.

Jonny says that he can partition the set (1,2,...,16) into two subsets S and T of the same size so that:

A: the sum of the numbers in S equals the sum of the numbers in T

B: the sum of the squares of the numbers in S equals the sum of the squares in T

c: the sum of the cubes of the numbers in S equals the sum of the cubes in T

Qc. Show/prove that Jonny is correct.

He then says he can partition the set (1,2,...,8) into two subsets S and T, not necessarily of the same size, so that:

A: the sum of the numbers in S equals the sum of the numbers in T

B: the sum of the squares of the numbers in S equals the sum of the squares in T

c: the sum of the cubes of the numbers in S equals the sum of the cubes in T

Qd: Explain why you do not believe him this time.

Qa. Find the next whole number n, above 4, for which the set (1,2,...,n) can be partitioned into two subsets S and T of the same size, with the sum of the numbers in S equal to the sum of the numbers in T. (this is farily simple as n can't be odd, and the sum of 1,...n can't be odd either)

Qb. Find all partitions in a with the ADDITIONAL property that the sum of the squares of the numbers in S equals the sum of the squares of the numbers in T.

Jonny says that he can partition the set (1,2,...,16) into two subsets S and T of the same size so that:

A: the sum of the numbers in S equals the sum of the numbers in T

B: the sum of the squares of the numbers in S equals the sum of the squares in T

c: the sum of the cubes of the numbers in S equals the sum of the cubes in T

Qc. Show/prove that Jonny is correct.

He then says he can partition the set (1,2,...,8) into two subsets S and T, not necessarily of the same size, so that:

A: the sum of the numbers in S equals the sum of the numbers in T

B: the sum of the squares of the numbers in S equals the sum of the squares in T

c: the sum of the cubes of the numbers in S equals the sum of the cubes in T

Qd: Explain why you do not believe him this time.