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There are two bottles of wine each of capacity 800 ml. These are to be shared among 4 friends so that each will get 400 ml. There is no measuring instrument to measure out the exact volume but there are two glasses that can measure out exactly 300 ml each. How can you pour out the wine so that each will get their 400 ml of wine? I tried but got into the point where 2 friends get 400 ml and another one gets 200 ml. However now in one of the bottles there is 600 ml of wine and then I cannot proceed further. No matter how I proceed from the beginning I seem get to a dead end.

This problem is from New Syllabus Additional Mathematics(7th Edition) by Lee Peng Yee

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  • $\begingroup$ since you only have two glasses, each holding less than the amount you expect someone to get, consider that people are going to drink their wine in shifts and/or from the bottle $\endgroup$ May 12, 2023 at 16:47
  • $\begingroup$ @KateGregory both ways are permitted as long as they get to have 400 ml ultimately. It is possible someone first has 300 ml and then at some other point 100 ml. $\endgroup$ May 12, 2023 at 16:49
  • $\begingroup$ is it a requirement that the person dispensing the wine should remain sober throughout ?-) $\endgroup$ May 13, 2023 at 17:55

2 Answers 2

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Ultimately this solution hinges upon getting the 2nd and 3rd drinkers to have 100ml so that we can easily fill up 2 wine glasses to get them the rest of their allotment and not worry about measuring for the 4th person. 100ml is achievable by pouring 2 300ml glass into a 800ml bottle (for 600 ml) and then trying to pour another 300ml glass into that bottle (which leaves a remainder of 100ml).

For simplicity, we'll label the two bottles A and B and the two glasses C and D. We'll also name the four friends John, Paul, Ringo, and George. We'll keep track of how much wine each container has, and how much each person has drank with each step.

Starting state:
A:8, B:8, C:0, D:0 | John: 0, Paul: 0, Ringo: 0, George: 0

1. Pour A into C and D (A:2, B:8, C:3, D:3 | J:0,P:0,R:0,G:0)
2. Have John drink from the bottle A (A:0, B:8, C:3, D:3 | J:2,P:0,R:0,G:0)
3. Pour C and D back into A (A:6, B:8, C:0, D:0 | J:2,P:0,R:0,G:0)
4. Pour B into C and D (A:6, B:2, C:3, D:3 | J:2,P:0,R:0,G:0)
5. Have John drink from the bottle B (A:6, B:0, C:3, D:3 | J:4,P:0,R:0,G:0)
6. Pour C into A (A:8, B:0, C:1, D:3 | J:4,P:0,R:0,G:0)
7. Have Paul drink C (A:8, B:0, C:0, D:3 | J:4,P:1,R:0,G:0)
8. Pour A into C (A:5, B:0, C:3, D:3 | J:4,P:1,R:0,G:0)
9. Pour C and D into B (A:5, B:6, C:0, D:0 | J:4,P:1,R:0,G:0)
10. Pour A into C (A:2, B:6, C:3, D:0 | J:4,P:1,R:0,G:0)
11. Pour C into B (A:2, B:8, C:1, D:0 | J:4,P:1,R:0,G:0)
12. Have Ringo drink C (A:2, B:8, C:0, D:0 | J:4,P:1,R:1,G:0)
13. Pour B into C and D (A:2, B:2, C:3, D:3 | J:4,P:1,R:1,G:0)
14. Have Paul drink C and Ringo drink D (A:2, B:2, C:0, D:0 | J:4,P:4,R:4,G:0)
15. Have George drink the remaining (A:0, B:0, C:0, D:0 | J:4,P:4,R:4,G:4)

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    $\begingroup$ It's absolutely critical that you have nothing but 2's and 3's left once you get down to ~1000ml left because that's where it starts to become impossible to get a 1. The mistake I made was not tracking the Beatles as individuals, that really makes the solution pop. $\endgroup$ May 12, 2023 at 22:02
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    $\begingroup$ Let's hope it is the same wine in both bottles because you are mixing the two :-) $\endgroup$
    – quarague
    May 13, 2023 at 8:25
  • $\begingroup$ And that no-one proposes a toast. $\endgroup$ May 13, 2023 at 15:43
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    $\begingroup$ You could totally do a toast. George drinks instead of John in step 5. Toast in step 13. $\endgroup$ May 13, 2023 at 17:44
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@JGibbbers' solution takes 22 steps and uses both glasses:

1. Pour A into C
2. Pour A into D
3. J drinks from bottle A
4. Pour C into A
5. Pour D into A
6. Pour B into C
7. Pour B into D
8. J drinks from bottle B
9. Pour C into A
10. P drinks C
11. Pour A into C
12. Pour C into B
13. Pour D into B
14. Pour A into C
15. Pour C into B
16. R drinks C
17. Pour B into C
18. Pour B into D
19. P drinks C
20. R drinks D
21. G drinks C
22. G drinks D

For 2 additional steps, you can avoid using the second glass completely! (because who wants to wash that many cups?)

Notation:

  • B1: amount in bottle 1 (800 ml capacity)
  • B2: amount in bottle 2 (800 ml capacity)
  • G: amount in glass (300 ml capacity)
  • A: amount drunk by person A
  • B: amount drunk by person B
  • C: amount drunk by person C
  • D: amount drunk by person D

B1: 800, B2: 800, G: 0 | A: 0, B: 0, C: 0, D: 0: starting (step 00)
B1: 500, B2: 800, G: 300 | A: 0, B: 0, C: 0, D: 0: B1 to G (step 01)
B1: 500, B2: 800, G: 0 | A: 300, B: 0, C: 0, D: 0: G to A (step 02)
B1: 200, B2: 800, G: 300 | A: 300, B: 0, C: 0, D: 0: B1 to G (step 03)
B1: 0, B2: 800, G: 300 | A: 300, B: 200, C: 0, D: 0: B1 to B (step 04)
B1: 300, B2: 800, G: 0 | A: 300, B: 200, C: 0, D: 0: G to B1 (step 05)
B1: 300, B2: 500, G: 300 | A: 300, B: 200, C: 0, D: 0: B2 to G (step 06)
B1: 600, B2: 500, G: 0 | A: 300, B: 200, C: 0, D: 0: G to B1 (step 07)
B1: 600, B2: 200, G: 300 | A: 300, B: 200, C: 0, D: 0: B2 to G (step 08)
B1: 800, B2: 200, G: 100 | A: 300, B: 200, C: 0, D: 0: G to B1 (step 09)
B1: 800, B2: 200, G: 0 | A: 400, B: 200, C: 0, D: 0: G to A (step 10)
B1: 500, B2: 200, G: 300 | A: 400, B: 200, C: 0, D: 0: B1 to G (step 11)
B1: 0, B2: 700, G: 300 | A: 400, B: 200, C: 0, D: 0: B1 to B2 (step 12)
B1: 300, B2: 700, G: 0 | A: 400, B: 200, C: 0, D: 0: G to B1 (step 13)
B1: 300, B2: 400, G: 300 | A: 400, B: 200, C: 0, D: 0: B2 to G (step 14)
B1: 600, B2: 400, G: 0 | A: 400, B: 200, C: 0, D: 0: G to B1 (step 15)
B1: 600, B2: 100, G: 300 | A: 400, B: 200, C: 0, D: 0: B2 to G (step 16)
B1: 800, B2: 100, G: 100 | A: 400, B: 200, C: 0, D: 0: G to B1 (step 17)
B1: 800, B2: 0, G: 100 | A: 400, B: 200, C: 100, D: 0: B2 to C (step 18)
B1: 800, B2: 0, G: 0 | A: 400, B: 200, C: 100, D: 100: G to D (step 19)
B1: 500, B2: 0, G: 300 | A: 400, B: 200, C: 100, D: 100: B1 to G (step 20)
B1: 500, B2: 0, G: 0 | A: 400, B: 200, C: 400, D: 100: G to C (step 21)
B1: 200, B2: 0, G: 300 | A: 400, B: 200, C: 400, D: 100: B1 to G (step 22)
B1: 0, B2: 0, G: 300 | A: 400, B: 400, C: 400, D: 100: B1 to B (step 23)
B1: 0, B2: 0, G: 0 | A: 400, B: 400, C: 400, D: 400: G to D (step 24)

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