Life is A Test 2: Test Harder (Quite A Bit Harder)

by Brony_of_Brody

Zecora Becomes An Accomplice

Having finally procured a series of medical stones, several powders, and a bottle of what smells suspiciously like barbeque sauce, the next step comes into play. Fluttershy explains to you that recently, Graham has come down with a bit of joint pain in his arms, and so a special brew needs to be concocted to restore vitality. Unfortunately, it's beyond Fluttershy's level of expertise to brew, and so, as is the case with anything involving potions, help from the zebra Zecora is required.

You drag all the ingredients to Zecora's hut, where she sets up a cauldron for the brew. The good news is that it's possible for her to make. The downside is that the brew requires a very specific ingredient to brew properly, and although you have a lot of it, not everything is the right quality. Specifically, the medical stones (they look a lot like kidney stones, but that's neither here nor there).

The recipe calls for stones that are twenty grams or more, and while the supplier has generously assured Fluttershy that every medical stone provided does weigh a whole number of grams, all of them look exactly the same: the only difference is in the weight. Fortunately, although Zecora seems to forgo the new-fangled things like the electronic scales that Canterlot chefs (and Twilight Sparkle whens she hits a 'mad scientist' phase) seem so very fond of, she does have at her disposal a set of old-fashioned balance scales, along with enough weights, weighing 1-40 grams, to go round.

As much as it would be easy to simply set a twenty-gram weight on one side and just use whatever stones are equal or heavier, Fluttershy seems insistent that every medical stone is weighed individually and recorded, so that the subsequent data can be used for next year. As much as the thought of doing this again next year makes you shudder, it's all part of your self-inflicted punishment, so you have to go along. So, what's the most efficient way to weigh all of these medical stones?

Author's Note:

1. A selection of medicinal rocks weighing between 1g and 40g.
2. All rocks weigh a whole number of grams.
3. You have a set of old-fashioned balance scales.

PUZZLE: What is the fewest number of weights needed to weigh all the stones?

Comments

[ANW]

1,2,4,8,16,32
That's 6 weight all together.

[Brony_of_Brody]

9371789
Nope. Try again!

[ANW]

We only need one. Literally.
The one weight. And only twice
We find the medical stone that weighs 1 gram.
Then use those to find the 2 gram stone.
Then using the 1 gram stone and two pound stone, we find the 3 gram stone.
And the pattern repeats. At one point, we'll have enough recorded stone to properly weigh each stone instead of looking for a specific amount.

[SisselSandvich]

ANW, your second solution assumes that every possible weight is used for a stone, which is not guaranteed.

For a few moments I was caught in the trap of ANW's first solution, but then I realized: weights can be added to both sides of the scale. So instead of 2 states for each weight (on the scale or off) that lead to the 1,2,4,8,16,32 solution, there are 3 states for each weight (unused, opposite the stone, or on the same side as the stone). So instead of powers of 2, we can use powers of 3: 1,3,9,27. For example, to test if a stone weights 8 grams, put it and the 1-gram weight opposite the 9-gram weight, so it would weigh 9-1=8 grams; to test for 16 put it, 9-gram, and 3-gram opposite 27-gram and 1-gram so it would weigh 27-9-3+1=16 grams, and so on.

Incidentally, I had thought of trying to reduce the number of weights further and take advantage of the fact that every stone weighs an integer number of grams by doubling every weight, figuring that if we discover a stone is more than 12 grams but less than 14 grams it must be 13 grams, but while it does cut out the 1-gram weight from ANW's solution, it doesn't actually remove any weights from mine - the lower three weights would double to 2, 6, and 18 grams, and this only adds up to 26 grams, so we would still need a fourth weight to get up to 40 grams. That in mind, I'm sticking with the 1,3,9,27 split.

[The Real Anonymous Pony]

9371824
If each stone weighed a different amount, then you would need no weights, just use the stones themselves. The 1 stone is lighter than the 2 stone. The 2 stone is lighter than the the 3 stone. Etc.

[ProfessorVenturer]

You don't need the scale; you can feel the difference in weight by holding both rocks in opposite hooves.