Hey, someone at work sent this to me and it’s breaking my damned head. Here’s the problem:
This doesn’t compute with simple algebra. I mean, I get:
J + (J - 8) + (J - 76) + (J - 187) = 1648. That works out to:
J = 479.75
P = 471.75
L = 403.75
D = 292.75
…and obviously, there are no fractions of votes allowed. (Insert your own joke about the 3/5 Compromise here.) Am I missing some vital, sneaky element of the original problem? I confirmed with her that it’s typed correctly.
Seems to me that if only 3 votes were for somebody else you’d get nice round numbers. There are lots of other possibilities, as long as you assume that other people (or “undecided”) are in the mix.
That would be too easy, I got the same answer as you OneCentStamp, with 4x-271+1648 and x being 479.75. I am going to keep staring at it a while to see if I figure out the trick.
Before anyone jumps down my throat to prove their intelligence, I meant 4x-271=1648
I’ve sent it round the office to see if anyone else has any luck, I can’t work it out.
1648/4=412, but 8+76+187=271 which doesn’t divide by 4…
Sorry, I mean it doesn’t divide by 3, which means you can’t distribute the 271 votes around three people evenly.
There are some election systems which use fractional votes, but you have to assume that they aren’t used in mayoral elections.
One possibility is that a single-transferrable vote system is being used. However, then solutions are not unique. One solution would be:
First count:
Joe - 505
Pete - 444
Looie - 381
Doug - 318
Second count:
Joe - 587
Pete - 550
Looie - 511
Third count:
Joe - 828
Pete - 820
For the first and second counts, many other solutions are possible. I think the solution that I have offered spreads the candidates in as close to an arithmetic progression as posible.
What’s wrong with CalMeacham’s idea that there were three votes not for any of these guys? Write-ins? Hanging chads? The problem is simple enough that you yourself see the problem - there is more info needed, and you don’t have it in the problem statement.
Slippery Joe ran against Sneaky Pete, Big Looie the Con, and Dragnet Doug for the Mayor’s job in a small town. Out of 1,648 total votes, Joe beat Pete by 8 votes, Looie by 76 votes and Doug by 187 votes. How many votes did each of these gentlemen receive?
Vote Variables: SJ=Slippery Joe, SP=Sneaky Pete, BLC=Big Louie and DD="dragnet Doug…OTH=Other/Undecided
SJ=SP+8
SJ=BLC+76
SJ=DD+187
Substituting, we get
or
2.1) 4SJ + 271 + OTH = 1648
or
2.2) 4SJ+OTH = 1648-271
or
We need OTH, the number of other/undecided votes…
But maybe the source she got it from had a typo in it. Hitting 8 instead of 5 is a common typo if you use the numeric keypad.
But that’s not a satisfying answer! 
Indeed: a satisfying answer is a unique answer, given the conditions. If 3 did not vote, or voted for other candidates, then why not 7, or 11, or 15, etc.? That’s why I don’r regard my answer as satisfactory, because it’s not unique.
Satisfying? :rolleyes: Algebra is useful, not satisfying.
We have a beautiful solution for SJ in terms of OTH. The competing restriction that OTH be a non-negative integer or zero, as well as the is the upper limit of 1648 total votes yields a family of possibilities. The non-uniqueness isn’t relevent…
Solve SJ = (1/4)*(1648-271-OTH) by substituiting OTH values… Begin with OTH=0, see if SJ is zero or a non-negative integer…At some point, you’ll run out of possibilities.
With people running named Slippery Joe, Sneaky Pete, and Big Looie the Con, how can you trust any of the numbers?
OTH = 4N + 3, where N is an integer.
There - solved.
This is as satisfying of an answer as you’re going to get without clarification from the person who gave you the problem statement.