“The following surprisingly difficult problem was given to my son's Math Circle”

Re:“The following surprisingly difficult problem was given t... by rileyjshaw on Tuesday October 21, 2014 @07:13AM (#8486797)

Here's how I solved it mathematically:

  Farmer 1 has 10 chickens.
  Farmer 2 has 16 chickens.
  Farmer 3 has 26 chickens.

  Price before lunch: x
  Price after lunch: y

  # of chickens sold before lunch by farmers 1, 2, 3: a, b, c

  Total earned by each farmer: $35

We're told that,

  x > y

and can logically deduce that,

  a > b > c

because otherwise, the farmers with less chickens would have no chance of making the same amount as the other farmer.*

From this information, we know that:

  ax + (10 - a)y = bx + (16 - b)y = cx + (26 - c)y = 35

Let's isolate the first and second farmers here:

  ax + (10 - a)y = bx + (16 - b)y
  ax - ay + 10y + bx - by + 16y = 0
  (a - b)(x - y) = 6y

We can do the same between farmers 1 and 3:

  (a - c)(x - y) = 16y

These two formulas yield,

  (a - b) = (3 / 8)(a - c)

Since a > b > c,

  (a - b) > 0
  (a - c) > 0

Since farmer 1 only has 10 chickens,

  a ≤ 10

Since you can't sell negative chickens,

  b ≥ 0
  c ≥ 0

And since the problem isn't very interesting if the farmers are allowed to sell half-chickens, a, b, and c (and the difference between them) are integers.

Given all of this, 0 ≤ (a - b), (a - c) ≤ 10. The only numbers that satisfy this and,

  (a - b) = (3 / 8)(a - c)

are,

  (a - b) = 3
  (a - c) = 8

Since c ≥ 0 and a ≤ 10, we have three triplets to consider:

  a = 10: {10, 7, 1}
  a = 9:  {9, 6, 1}
  a = 8:  {8, 5, 0}

We can find the relationship between x and y from an earlier equation:

  (a - b)(x - y) = 6y
  3(x - y) = 6y
  3x = 9y
  x = 3y

So the farmers reduced their price to a third of the original price during the afternoon. What a deal!

We've got a few equations that look like,

  ax + (10 - a)y = 35

Which we can now simplify to,

  2ay + 10y = 35

By plugging [10, 9, 8] into the above formula, the only value that gives us a proper dollar amount for y is a = 9.

So...

  y = $1.25
  x = $4.25

Reading through the G+ comments it looks like someone beat me to it, but I figured I'd share my solution anyway.

*This is assuming that they didn't decide to "sell" their chickens for $0 in the afternoon, which is probably a safe bet.

Edit: add intermediate steps for clarity

[ Parent | on HN ]

Re:“The following surprisingly difficult problem was given t... by Tloewald on Tuesday October 21, 2014 @09:39AM (#8487229)

Setting aside the trivial arithmetic error, I think a simpler attack approach comes from the first two farmers collectively having 26 chickens and making twice as much money as the third, giving you a simpler starting point.

[ Parent | on HN ]

Re:“The following surprisingly difficult problem was given t... by _paulc on Tuesday October 21, 2014 @07:37AM (#8486840)

Except that I get a different answer if you do a very simple bruteforce:

    def check(n,am,pm,total):
        for x in range(1,n):
            if (am * x + pm * (n-x)) == total:
                return "AM: %d @ %d / PM %d @ %d" % (x,am,n-x,pm)
        return None


    def solve(total,birds):
        for am in range(1,total):
            for pm in range(1,am):
                result = list(map(lambda n:check(n,am,pm,total),birds))
                if all(result):
                    return result

    print(solve(3500,[10,16,26]))

[ Parent | on HN ]

Re:“The following surprisingly difficult problem was given t... by botzi2001 on Tuesday October 21, 2014 @07:29AM (#8486822)

One small mistake at the end: x = $1.25 y = $3.75

[ Parent | on HN ]

Re:“The following surprisingly difficult problem was given t... by rileyjshaw on Tuesday October 21, 2014 @03:18PM (#8489232)

o______o

oh my. this is why one shouldn't do math at 4am.

:s/4.25/3.75

[ Parent | on HN ]