# Willy's mistake

darkmage
Posers and Puzzles 15 May '05 07:54
1. 15 May '05 07:54
The set (1,2,3,4) can be partioned into two subsets (1,4) and (2,3) of the same size. 1+4=2+3
a)find the next whole number n, above 4, for which the set (1,2,...n) can be paritioned into two subsets S and T of the same size, with the sum of S and T the same.

b)find all partition 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.

Terri says she can partition the set (1,2...16) into subsets S and T of the same size so that:
- the sum of the numbers in S equals the sum of T
-the sum of the squares of the number in S equals the sum of the squares of the numbers in T.
-the sum of the cubes of the numbers in S equals the sum of the cubes of the numbers in T.

c) show terri is right.

Willy says she can partition the set (1,2...8) into subsets S and T, not necessarily the same size so that:
- the sum of the numbers in S equals the sum of T
-the sum of the squares of the number in S equals the sum of the squares of the numbers in T.
-the sum of the cubes of the numbers in S equals the sum of the cubes of the numbers in T.

d)show why you dont belive willy
2. 15 May '05 08:52
Originally posted by darkmage
The set (1,2,3,4) can be partioned into two subsets (1,4) and (2,3) of the same size. 1+4=2+3
a)find the next whole number n, above 4, for which the set (1,2,...n) can be paritioned into two subsets S and T of the same size, with the sum of S and T the same.

b)find all partition in a) with the additional property that the sum of the squares of the numbe ...[text shortened]... umbers in S equals the sum of the cubes of the numbers in T.

d)show why you dont belive willy
It appears as though you and phgao are both trying to cheat on the same 2005 Intermediate Maths Challenge, whatever that is. I am just basing this on 12122000's post in phgao "sets" thread, but i do find it quite interesting that your question matches his more or less exactly.
3. 15 May '05 12:49
not "cheat" getting help
4. 16 May '05 01:58
Originally posted by darkmage
The set (1,2,3,4) can be partioned into two subsets (1,4) and (2,3) of the same size. 1+4=2+3
a)find the next whole number n, above 4, for which the set (1,2,...n) can be paritioned into two subsets S and T of the same size, with the sum of S and T the same.

b)find all partition in a) with the additional property that the sum of the squares of the numbe ...[text shortened]... umbers in S equals the sum of the cubes of the numbers in T.

d)show why you dont belive willy
I can solve part "a". You can take any set of consecutive numbers that has a cardinality divisible by 4. So, the next set would be "1 2 3 4 5 6 7 8". Notice that if you reverse the numbers, and add 1 with 8, 2 with 7, 3 with 6, and 4 with 5, you get 9 for each addition. Now, you know your pairs. Because you have 4 subsets of 2, you can group any two subsets of 2 you want, and call them S, and the other two subsets are T. So, you have set S = (1, 8, 2, 7), and T = (3, 6, 4, 5). If you had an original set of twelve, you would use the same trick of reversing the order, and then group 3 subsets of two, to get S, and T would be the rest.
ðŸ™‚
5. 16 May '05 05:41
Originally posted by darkmage
Willy says she can...
Willy is a girl???
6. 16 May '05 11:54