Week 46: PUZZLE PARTY

Inspired by the success of Florida's 2000 ballot system at weeding out low-IQ voters, Ohio's 2004 election board employed the following punchcard ballot for the 2004 election.  To make your vote count, place your vote for the Puzzle party.

			Republican + Democrat = Green Democrat * 3 = Libertarian Libertarian/Constitution = Reform Green - Constitution = Puzzle Reform * 3 = Democrat Puzzle/Green = Republican/Puzzle
1	2	3	4	5	6	7	8	9	10
11	12	13	14	15	16	17	18	19	20
21	22	23	24	25	26	27	28	29	30
31	32	33	34	35	36	37	38	39	40
41	42	43	44	45	46	47	48	49	50

ANSWER: 18

One way to solve it was as follows.  Start by solving for the Constitution party:

Democrat * 3 = Libertarian
Reform * 3 = Democrat
Libertarian/Constitution = Reform
so...
Reform * 9 = Libertarian
Constitution = (Reform * 9)/Reform
Constitution = 9

Since all answers have to be integers < 50, the possibilities for Libertarian/Democrat/Reform are:

Libertarian = 9, Democrat = 3, Reform = 1
Libertarian = 18, Democrat = 6, Reform = 2
Libertarian = 27, Democrat = 9, Reform = 3
Libertarian = 36, Democrat = 12, Reform = 4
Libertarian = 45, Democrat = 15, Reform = 5

We can solve for Green in terms of Democrat:

Green - 9 = Puzzle
Green = Republican + Democrat
Puzzle² = Green * Republican
(Green - 9)² = Green * (Green - Democrat)
Green² - 18*Green + 81 = Green² - Democrat*Green
(18-Democrat)*Green = 81

So the possibilities for Green are:

if Democrat = 3, Green = 5.4
if Democrat = 6, Green = 6.75
if Democrat = 9, Green = 9
if Democrat = 12, Green = 13.5
if Democrat = 15, Green = 27

Only the last possibility is valid, so...

Constitution = 9, Libertarian = 45, Democrat = 15, Reform = 5, Green = 27, Republican = 12, and Puzzle = 18.