How many smurfs are there?

Papa Smurf gathered some berries from the forest to the village and smurfs ate all those berries:

• The smurf who ate the most actually ate one-fourth of the berries eaten by the rest of the smurfs.
• The smurf who ate the third-most actually ate one-ninth of the berries eaten by the rest of the smurfs
• The smurf who ate the least actually ate one-tenth of the berries eaten by the rest of the smurfs.

How many smurfs are there?

Reference: Bilim ve Teknik Dergisi 2018-08

Let $N$ be the number of smurfs.

The information we are given implies that one smurf ate $1/5$ of the total berries, and that each of the other $N-1$ smurfs ate at least $1/11$ of the total berries. This means we must have $$\frac{1}{5}+(N-1)\frac{1}{11} \leq 1,$$ and so $N\leq 9$.

The the smurf who ate the second-most at strictly less than $\frac{1}{5}$ of the total, and each of the $N-2$ smurfs (other than those who at the most and second most) ate at most $\frac{1}{10}$ of the total. This means we must have $$\frac{1}{5}+\frac{1}{5}+(N-2)\frac{1}{10}>1,$$ and so $N\geq 9$.

Taken together, these bounds imply there must be $9$ smurfs.

• Curious, why 1/5? – mascoj Aug 5 '18 at 20:08
• Let T be the total number of berries and S be number that the hungriest smurf ate. From the first clue we know that S = .25(T-S). Solving for S gives T/5. – jamisans Aug 5 '18 at 21:49
• welcome back @juliantosen, good answer! – Oray Aug 6 '18 at 5:32

I have an answer that works, but no proof of uniqueness yet.

There are 9 smurfs. One way this works is for there to be 1100 berries. The number of berries that each smurf eats is 220, 135, 110, 109, 108, 107, 106, 105, 100. The numbers sum to 1100. The greatest value, 220, is one-fourth of the remaining sum (1100-220=880). The third greatest value, 110, is one-ninth of the remaining sum (1100-110=990). The smallest value, 100, is one-tenth of the remaining sum (1100-100=1000).

"The smurf who ate the most actually ate one-fourth of the berries eaten by the rest of the smurfs." -> This sentence implies that the smurth who ate the most ate 1/6 of the total berries (so there are at least 6 smurths).

"The smurf who ate the third-most actually ate one-ninth of the berries eaten by the rest of the smurfs" -> So the smurth who ate the third-most actually ate one-tenth of the total berries. So someone ate between 1/6 and 1/10.

"The smurf who ate the least actually ate one-tenth of the berries eaten by the rest of the smurfs" -> So everyone ate less than 1/11.

Let x the quantity eaten by the second-most (1/6 > x > 1/10). Let y the quantity eaten by the N-4 others smurfs ( (N-4)/11 < y < (N-4)/10 )

So the total quantity is x + y + 1/6 + 1/10 + 1/11 And so, we search N so that

1/6 + (N-4)/10 + 1/6 + 1/10 + 1/11 > 1 > 1/10 + (N-4)/11 + 1/6 + 1/10 + 1/11

So, N=9. There are 9 smurfs.

• Some errors: First sentence should be 1/5 not 1/6 (he ate 1/4 of the rest) and third paragraph: One Smurf ate 1/11 - all other Smurfs ate more than 1/11 – Falco Aug 6 '18 at 11:06