[rookie1ja]

Weighing II. - Back to the Water and Weighing Puzzles
A genuine gummy drop bear has a mass of 10 grams, while an imitation gummy drop bear has a mass of 9 grams. Spike has 7 cartons of gummy drop bears, 4 of which contain real gummy drop bears and the others – imitation.
Using a scale only once and the minimum number of gummy drop bears, how can Spike determine which cartons contain real gummy drop bears?

Spoiler for Solution:

Weighing II. - solution
Spike uses 51 gummy drop bears: from the 7 boxes he takes respectively 0, 1, 2, 4, 7, 13, and 24 bears.
The notion is that each box of imitation bears will subtract its number of bears from the total "ideal" weight of 510 grams (1 gram of missing weight per bear), so Spike weighs the bears, subtracts the result from 510 to obtain a number N, and finds the unique combination of 3 numbers from the above list (since there are 3 "imitation" boxes) that sum to N.
The trick is for the sums of all triples selected from the set S of numbers of bears to be unique. To accomplish this, I put numbers into S one at a time in ascending order, starting with the obvious choice, 0. (Why is this obvious? If I'd started with k > 0, then I could have improved on the resulting solution by subtracting k from each number) Each new number obviously had to be greater than any previous, because otherwise sums are not unique, but also the sums it made when paired with any previous number had to be distinct from all previous pairs (otherwise when this pair is combined with a third number you can't distinguish it from the other pair)--except for the last box, where we can ignore this point. And most obviously all the new triples had to be distinct from any old triples; it was easy to find what the new triples were by adding the newest number to each old sum of pairs.
Now, in case you're curious, the possible weight deficits and their unique decompositions are:
3 = 0 + 1 + 2
5 = 0 + 1 + 4
6 = 0 + 2 + 4
7 = 1 + 2 + 4
8 = 0 + 1 + 7
9 = 0 + 2 + 7
10 = 1 + 2 + 7
11 = 0 + 4 + 7
12 = 1 + 4 + 7
13 = 2 + 4 + 7
14 = 0 + 1 + 13
15 = 0 + 2 + 13
16 = 1 + 2 + 13
17 = 0 + 4 + 13
18 = 1 + 4 + 13
19 = 2 + 4 + 13
20 = 0 + 7 + 13
21 = 1 + 7 + 13
22 = 2 + 7 + 13
24 = 4 + 7 + 13
25 = 0 + 1 + 24
26 = 0 + 2 + 24
27 = 1 + 2 + 24
28 = 0 + 4 + 24
29 = 1 + 4 + 24
30 = 2 + 4 + 24
31 = 0 + 7 + 24
32 = 1 + 7 + 24
33 = 2 + 7 + 24
35 = 4 + 7 + 24
37 = 0 + 13 + 24
38 = 1 + 13 + 24
39 = 2 + 13 + 24
41 = 4 + 13 + 24
44 = 7 + 13 + 24
Note that there had to be (7 choose 3) distinct values; they end up ranging from 3 to 44 inclusive with 7 numbers missing: 4, 23, 34, 36, 40, 42, and 43.

Spoiler for old wording:

Real gummy drop bears have a mass of 10 grams, while imitation gummy drop bears have a mass of 9 grams. Spike has 7 cartons of gummy drop bears, 4 of which contain real gummy drop bears, the others imitation. Using a scale only once and the minimum number of gummy drop bears, how can Spike determine which cartons contain real gummy drop bears?

Edit: scales (for this case) = weighing machine (like in science labs, so no pair of scales) and the plates are big enough for all possible weighings
Edit: each carton contains 50 bears

[kola]

This sequence uses minimum gummy bears.

0, 1, 2, 3, 6, 11, 20

This covers all unique sums from 3 to 37 ( 35 unique values = 7 choose 3).

[kola]

Sorry, the previous sequence posted by me does not work. You were right.

[atrophic]

Quote

Each new number obviously had to be greater than any previous, because otherwise sums are not unique, but also the sums it made when paired with any previous number had to be distinct from all previous pairs (otherwise when this pair is combined with a third number you can't distinguish it from the other pair)--except for the last box, where we can ignore this point.

While the solution is correct, the logic quoted above in attaining it is not. By that logic the 6th bag (13) could be 10 instead as no two numbers before it add up to 10. However, 10+1 = 7+4. Therefore, you also have to check to be sure that the number plus any previous number does not sum to the same as any two previous numbers. That will get you a little farther (to 13 appropriately). But even with that additional caveat you'll find problems. 21 would be the next available number using those rules, but 0+1+21 = 2+7+13. So you'd also have to add the restriction that the new number plus any 2 previous numbers can't sum to the sum of any three previous numbers, which is just stating the problem of unique triples over again.

So... here's the logic you have to use to solve it: Start with 0, 1, 2, 4 and thereafter the next number in the series = the sum of the last three numbers. The sum of the last three numbers in the series is the highest possible sum to that point, and the new number must at a minimum add 0 and 1, thereby guaranteeing unique triples.

[therealdonquixote]

I have one complaint about this puzzle -
The type of scale was not defined in the puzzle. Typically the type of scale used in these weight/weighing puzzles is a balance, where some amount of billiard balls are weighed against another amount of billiard balls. However the solution to this puzzle, points to a scale that is not a simple balance. For this puzzle, the scale has to be either a balance with weights, or your typical scale used in science labs.

Other than that small vague area of the scale, its a perfectly delightful puzzle that requires great ingenuity to solve. Bravo.

* Totally nitpicking on the size of the plate, we all know these puzzles deal in ideal (unrealistic) circumstances. Like physics problems that you get before learning how to account for friction. However... the scale would need a rather large weighing plate to allow for 6 sets (the seventh has no bears at all) to be placed on it without getting the sets all mixed together. If spike had to use little plastic baggies to keep the bears seperate, then he would be forced to weigh the the empty baggies first, thus using the scale twice.

[rookie1ja]

Quote

I have one complaint about this puzzle -
The type of scale was not defined in the puzzle. Typically the type of scale used in these weight/weighing puzzles is a balance, where some amount of billiard balls are weighed against another amount of billiard balls. However the solution to this puzzle, points to a scale that is not a simple balance. For this puzzle, the scale has to be either a balance with weights, or your typical scale used in science labs.

right, I have edited the puzzle

[tshurtz]

Quote

to be placed on it without getting the sets all mixed together. If spike had to use little plastic baggies to keep the bears seperate, then he would be forced to weigh the the empty baggies first, thus using the scale twice.

the beauty is they can mix
you can eat them afterwards if you want

[aishi_khurana]

The trick is for the sums of all triples selected from the set S of numbers of bears to be unique.
could you please elaborate on this line

[spoxjox]

Quote

Weighing II. - Back to the Water and Weighing Puzzles
Real gummy drop bears have a mass of 10 grams, while imitation gummy drop bears have a mass of 9 grams. Spike has 7 cartons of gummy drop bears, 4 of which contain real gummy drop bears, the others imitation. Using a scale only once and the minimum number of gummy drop bears, how can Spike determine which cartons contain real gummy drop bears?

Edit: scales (for this case) = weighing machine (like in science labs, so no pair of scales) and the plates are big enough for all possible weighings

Spoiler for Solution:

Weighing II. - solution
Spike uses 51 gummy drop bears: from the 7 boxes he takes respectively 0, 1, 2, 4, 7, 13, and 24 bears.
The notion is that each box of imitation bears will subtract its number of bears from the total "ideal" weight of 510 grams (1 gram of missing weight per bear), so Spike weighs the bears, subtracts the result from 510 to obtain a number N, and finds the unique combination of 3 numbers from the above list (since there are 3 "imitation" boxes) that sum to N.
The trick is for the sums of all triples selected from the set S of numbers of bears to be unique. To accomplish this, I put numbers into S one at a time in ascending order, starting with the obvious choice, 0. (Why is this obvious? If I'd started with k > 0, then I could have improved on the resulting solution by subtracting k from each number) Each new number obviously had to be greater than any previous, because otherwise sums are not unique, but also the sums it made when paired with any previous number had to be distinct from all previous pairs (otherwise when this pair is combined with a third number you can't distinguish it from the other pair)--except for the last box, where we can ignore this point. And most obviously all the new triples had to be distinct from any old triples; it was easy to find what the new triples were by adding the newest number to each old sum of pairs.
Now, in case you're curious, the possible weight deficits and their unique decompositions are:
3 = 0 + 1 + 2
5 = 0 + 1 + 4
6 = 0 + 2 + 4
7 = 1 + 2 + 4
8 = 0 + 1 + 7
9 = 0 + 2 + 7
10 = 1 + 2 + 7
11 = 0 + 4 + 7
12 = 1 + 4 + 7
13 = 2 + 4 + 7
14 = 0 + 1 + 13
15 = 0 + 2 + 13
16 = 1 + 2 + 13
17 = 0 + 4 + 13
18 = 1 + 4 + 13
19 = 2 + 4 + 13
20 = 0 + 7 + 13
21 = 1 + 7 + 13
22 = 2 + 7 + 13
24 = 4 + 7 + 13
25 = 0 + 1 + 24
26 = 0 + 2 + 24
27 = 1 + 2 + 24
28 = 0 + 4 + 24
29 = 1 + 4 + 24
30 = 2 + 4 + 24
31 = 0 + 7 + 24
32 = 1 + 7 + 24
33 = 2 + 7 + 24
35 = 4 + 7 + 24
37 = 0 + 13 + 24
38 = 1 + 13 + 24
39 = 2 + 13 + 24
41 = 4 + 13 + 24
44 = 7 + 13 + 24
that there had to be (7 choose 3) distinct values; they end up ranging from 3 to 44 inclusive with 7 numbers missing: 4, 23, 34, 36, 40, 42, and 43.

This answer gives a total required of 0 + 1 + 2 + 4 + 7 + 13 + 24 = 51 bears. This is not a minimum, however, so it is wrong. Since we know that there are exactly four of the seven boxes that contain fake bears, we can use this fact to get a better number sequence.

Spoiler for ...:

0
1
2
3
4
8
15

for a total of 33 bears. Any group of four of these numbers yields, I believe, a unique sum.

[rookie1ja]

Quote

Quote

Weighing II. - Back to the Water and Weighing Puzzles
Real gummy drop bears have a mass of 10 grams, while imitation gummy drop bears have a mass of 9 grams. Spike has 7 cartons of gummy drop bears, 4 of which contain real gummy drop bears, the others imitation. Using a scale only once and the minimum number of gummy drop bears, how can Spike determine which cartons contain real gummy drop bears?

Edit: scales (for this case) = weighing machine (like in science labs, so no pair of scales) and the plates are big enough for all possible weighings

Spoiler for Solution:

Weighing II. - solution
Spike uses 51 gummy drop bears: from the 7 boxes he takes respectively 0, 1, 2, 4, 7, 13, and 24 bears.
The notion is that each box of imitation bears will subtract its number of bears from the total "ideal" weight of 510 grams (1 gram of missing weight per bear), so Spike weighs the bears, subtracts the result from 510 to obtain a number N, and finds the unique combination of 3 numbers from the above list (since there are 3 "imitation" boxes) that sum to N.
The trick is for the sums of all triples selected from the set S of numbers of bears to be unique. To accomplish this, I put numbers into S one at a time in ascending order, starting with the obvious choice, 0. (Why is this obvious? If I'd started with k > 0, then I could have improved on the resulting solution by subtracting k from each number) Each new number obviously had to be greater than any previous, because otherwise sums are not unique, but also the sums it made when paired with any previous number had to be distinct from all previous pairs (otherwise when this pair is combined with a third number you can't distinguish it from the other pair)--except for the last box, where we can ignore this point. And most obviously all the new triples had to be distinct from any old triples; it was easy to find what the new triples were by adding the newest number to each old sum of pairs.
Now, in case you're curious, the possible weight deficits and their unique decompositions are:
3 = 0 + 1 + 2
5 = 0 + 1 + 4
6 = 0 + 2 + 4
7 = 1 + 2 + 4
8 = 0 + 1 + 7
9 = 0 + 2 + 7
10 = 1 + 2 + 7
11 = 0 + 4 + 7
12 = 1 + 4 + 7
13 = 2 + 4 + 7
14 = 0 + 1 + 13
15 = 0 + 2 + 13
16 = 1 + 2 + 13
17 = 0 + 4 + 13
18 = 1 + 4 + 13
19 = 2 + 4 + 13
20 = 0 + 7 + 13
21 = 1 + 7 + 13
22 = 2 + 7 + 13
24 = 4 + 7 + 13
25 = 0 + 1 + 24
26 = 0 + 2 + 24
27 = 1 + 2 + 24
28 = 0 + 4 + 24
29 = 1 + 4 + 24
30 = 2 + 4 + 24
31 = 0 + 7 + 24
32 = 1 + 7 + 24
33 = 2 + 7 + 24
35 = 4 + 7 + 24
37 = 0 + 13 + 24
38 = 1 + 13 + 24
39 = 2 + 13 + 24
41 = 4 + 13 + 24
44 = 7 + 13 + 24
that there had to be (7 choose 3) distinct values; they end up ranging from 3 to 44 inclusive with 7 numbers missing: 4, 23, 34, 36, 40, 42, and 43.

This answer gives a total required of 0 + 1 + 2 + 4 + 7 + 13 + 24 = 51 bears. This is not a minimum, however, so it is wrong. Since we know that there are exactly four of the seven boxes that contain fake bears, we can use this fact to get a better number sequence.

Spoiler for ...:

0
1
2
3
4
8
15

for a total of 33 bears. Any group of four of these numbers yields, I believe, a unique sum.

I don't think so ... what about 15
8+4+2+1=15
8+4+3+0=15