# The Witch Invites

Halloween is just around the corner and the witch is planning a little party for the 100 villagers of the nearby village. Since the villagers burned down her house again and again during the last few years she would like to use this chance to make peace with them.

At least that is what she told the villagers.

Of course an evil witch would not simply forgive being attacked and so she collected the essences of animals, to be exact the essences of bears, hawks, and spiders, during the past year. Her plan is to use these essences to concoct potions which would transform the drinker into a creature with characteristics of all 3 animals.

Of course there are rules for these potions:

• The witch must concoct each potion on its own so 100 potions in total.
• For the concocting process all 3 essences are needed in a specific order, therefore each potion must contain at least a small amount of each essence.
• Each potion must must be concocted of the same amount of combined essences to get potions of equal strength. Otherwise the villagers who drank a stronger potion would be able to cancel the transformation of those who drank weaker potions.
• So the villagers would never find happiness again no two villagers should have the same transformation. Therefore each potion needs to be concocted using a different combination of amounts of essences.

Especially her last rule is a big problem for the witch.

She is only in posession of 4 magical cauldrons. One to contain each essence and one to concoct potions.

Unless the essences are in a complete potion or in one of the cauldrons they will dissipate in a matter of seconds so the witch can't measure the amount of essences for each potion beforehand.

Furthermore the concocting process is extremely complex and if she doesn't constantly stir it the incomplete potion would explode in her face.

For these reasons she can only use one ladle per essence to measure the needed amount for a potion.

How many different sized ladles does the witch need to concoct all 100 potions?

She would need

Explained:

We'll use EP (essence portion) as a unit.
She needs same strength potions for each villager, meaning the amount of EPs in each potion is equal. To get 100 different potions we need a number which can be summed up to using 3 parameters in 100 different ways.
I don't remember the proper math, but a small test on the computer showed that 16 is the smallest available, having 105 permutations.
Since the different compositions add up to 16, and we will have at least 1EP of each, she can ladel at most 14 EP of one.
Needing only 100 potions we can eliminate the 3 permutations using 14 (1-1-14, 1-14-1, 14-1-1), and we're left with 13 ladels.

• You can get 105 by doing $\binom{15}{2}$. That means the number of ways you can take 2 objects in a collection of 15. This is correct because in order to divide 16 objects, imagine they are all in a line. Because there is at least 1 item in a group we already take the first and put it aside, then point at 2 others and say that every item left of the first I pointed to belongs to group one and everything left of the second I pointed to belongs to group two and the rest in group three
– Ivo
Oct 29, 2015 at 10:16
• @ivobeckers, that sounds good :-)
– JNF
Oct 29, 2015 at 10:33
• Bonus for saving a ladle. Oct 29, 2015 at 20:02

She will only need 3 ladles of the same size, 1 for each essence(since she can only use one ladle per essence)

Because:

She can put 10 measures of the first essence(B), then 10 measures of the second(H) and then 10 of the third(S) to make 1 potion, then she can put 11B measures, 10H measures and 10S. This gives us 2 potions of equal strength(30 measures), but different combination of amounts.
Putting 10H first and then 10B and 10S will also produce a different potion, so she has plenty of choice and she only needs the ladle(s) of same size

• There may be an ambiguity in the wording of the question in regards to "can only use one ladle per essence". I'm interpreting it as "the act of ladling can only be performed once per essence". In which case, ladling 1 measure of essence B into the cauldron ten times in a row wouldn't be allowed. Oct 29, 2015 at 12:44

0, as using a ladle would mean the essences would dissipate:

Unless the essences are in a complete potion or in one of the cauldrons they will dissipate in a matter of seconds so the witch can't measure the amount of essences for each potion beforehand.

• If the cauldrons are close together, it should be simple to ladle essence out of one and into another in less than a second, in which case no dissipation should occur. Oct 29, 2015 at 12:47