All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
Pumpkins 6 (Posted on 2016-01-22) Difficulty: 4 of 5
Much like the third Pumpkins puzzle six pumpkins, each having a different weight, are weighed two at a time in all 15 sets of two. This time exactly four pairs of duplicate values occurred. Those four values are 60, 100, 110, and 120 pounds.

How much did each individual pumpkin weigh?

See The Solution Submitted by Brian Smith    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Analytical solution | Comment 2 of 11 |
Let the pumpkins weigh a,b,c,d,e,f in order of increasing weight.

1) a+b and a+c are both less than all other pairs, so 60 must = a+d = b+c.  Therefore, d = 60-a and c = 60-b

2) Similarly, 120 must = c+f = d+e.
    So, f = 120-c = 60+b, and e = 120-d = 60+a

The 6 pumpkins (in increasing order) weigh a,b,60-b,60-a,60+a,60+b

3) b<(60-b), so b < 30

4) At least 8 of the 15 weighings are divisible by 10.  8 is more than half of 15, so all of the pumpkins must have weights that are multiples of 5. 

5) 60+a+b = a+f = b+e, so that is one of the duplicate weighings, and it must be either 100 or 110
 5a) if 60+a+b = 110, then a+b = 50
     But there are no values of a+b=50 such that a < b < 30 and a and b are multiples of 5.
 5b) Therefore, 60+a+b = 100, so a+b = 40
     There is only one possiblity of a+b=40 such that a < b < 30 and a and b are multiples of 5.
     Namely a = 15, b = 25
     
THEREFORE, the pumpkins can only be (15,25,35,45,75,85).

Checking: 110 = 25+85 = 35+75, so everything works

  Posted by Steve Herman on 2016-01-23 10:21:15
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (2)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2020 by Animus Pactum Consulting. All rights reserved. Privacy Information