A wise man has three clay vessels: one of 12 liters, one of 8 liters and one of 5 liters. The 12 liter vessel is filled with sacred oil. You must divide the oil into two exactly equal parts of 6 liters using only these three vessels. How do you do it?
Problem of the Bakhshali Manuscript, 3rd-4th century
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The three vessels of the wise man (ancient India)
Hints
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- The strategy first builds a 1 liter "useful residue" in the large pot (step 5). From there, when completing the 5 liter (step 6), there are exactly 6 liters left in the 8; When emptying the 5 into the 12 (step 7), there are also 6 left in the 12.
- Status notation: (12L,8L,5L).
- Conclusion: the oil is divided into two equal parts of 6 liters using only permitted transfers.
Solution
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Answer: Yes, it is achieved in 7 transfers.
State notation:
$$ (12L,8L,5L). $$
Initial state:
$$ (12,0,0). $$
Sequence:
- $(12,0,0)\to(4,8,0)$
- $(4,8,0)\to(4,3,5)$
- $(4,3,5)\to(9,3,0)$
- $(9,3,0)\to(9,0,3)$
- $(9,0,3)\to(1,8,3)$
- $(1,8,3)\to(1,6,5)$
- $(1,6,5)\to(6,6,0)$
Final state:
$$ (6,6,0). $$
Why this route works
The strategy first builds a 1 liter "useful residue" in the large pot (step 5). From there, when completing the 5 liter (step 6), there are exactly 6 liters left in the 8; When emptying the 5 into the 12 (step 7), there are also 6 left in the 12.
Conclusion: The oil is divided into two equal parts of 6 liters using only permitted transfers.
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