# The weight of the Seven Dwarfs

Snow White offered the Seven Dwarfs a weighing machine. One after the other, they went on the machine and they wrote their weights on a piece of paper. Then they gave it to Snow White, but without writing their names:

22 kilograms

14 kgs

16 kgs

11 kgs

17 kgs

24 kgs

19 kgs

Feeling playful, they also went two by two on the machine. Except Grumpy, who didn't want to join the fun, obviously. Once they were done, they tell Snow White that:

• Doc and Sneezy were on the machine at the same time;

• Dopey and Happy were too;

• Bashful and Sleepy were the last two dwarfs to go together on the machine.

Altogether, they say to Snow White that the weight shown was the same each time. Snow White then says: "Don't tell me anything more, I know how much Grumpy weighs."

And you, do you know how much Grumpy weighs?

Grumpy weighs

24 kg

because

the others can be paired to add to 33 kg, but 24 kg is the odd weight out.
(11+22), (19+14), (16+17)

And to make sure

24, as the largest weight, can't be one of the pairs, because if you add it to the smallest, you get 35 kg. Since 22 has no pair to make 35, it would be the odd weight out, but the remaining four sum to 66, which is too small to make 2 35s.

If 24 were paired with 14 (the next-smallest option) you get 38 kg. But the sum of the 4 largest remaining weights is 74 kg, which is not enough to make two more 38 kg pairs. So 24 must be the odd weight out (assuming the puzzle has a solution, which the first part shows it does!).

• +1. The only answer to provide both the pairing and the reason =) Jul 26, 2016 at 16:28
• And I'll accept this one for the same reason, even if others have been providing good ones! Jul 26, 2016 at 17:30
• +1 for "And to make sure" part. I got the "because" part but the second explanation was really nice. Jul 28, 2016 at 4:29

Grumpy weighs

24 kilograms

How I worked it out

Add all of the weights together = 123 kg (note that this is divisible by 3). If we take grumpy's weight out, the total must still be divisible by 3, as there are 3 pairs of dwarves that were weighed. So Grumpy's weight MUST be divisible by 3. The only weight divisible by 3 is 24

• much neater explanation Jul 26, 2016 at 18:40
• Another good explanation. Jul 28, 2016 at 4:30

I think he is

24 kgs.

Reasoning:

the sum of their weights is 123kg. As the three groups of two measured the same, the total weight of those six is divisible by 3. Which means, Grumpy's weight must give the save remainder by 3 as 123. The only such number is 24.

Grumpy weights:

24 kilograms

Because

all the other numbers can be added in pairs to sum 33
22 + 11 ; 16 + 17; 19 + 14

I think Grumpy weighs

24kg
Since all the others should weigh equally pairs combining 11+22 we get 33 17+ 16 also yields the same result and so does 14+19

• the same answer has already been given with the same explanation Jul 26, 2016 at 18:42