Game of Dark Mages
Once upon a time, in the magic land called Mathematica, there were 4 dark mages, who strove to be greatest of them all. After numerous battles, however, their powers were too evenly matched with no decisive victory for any of them. Thus, to avoid futile bloodshed, all of them congregated in the chamber of darkness, where the darkest magic inside would destine only one person to live.
Suddenly, the four small clay oil lamps were magically lit on the four walls with the "fire of life" of the four mages individually, and they did not know which light belonged to them. Each lamp would initially contain 2 spoons of oil equally as its fuel.
Then the four jugs of oil magically appeared before the hands of each mage. To start the ritual, all of them would separately walk around the room, saying the divine incantation. After one circling, each lamp would deplete 1 spoon of oil, and if the oil ran out, the fire would burn out, leading to an end of one mage's life. Thus, within each round, each mage would pour exactly 1 spoon of oil into one lamp only to maintain the light. However, the room was too dim to see which lamps other mages had filled, and they could not estimate the amount of oil in any lamp due to dark clay color. Extending their own lives depended on pure chances, and the fire could not be extinguished by any other means.
After some rounds, one light was out, and one mage simultaneously dropped dead before vanishing magically along with the oil jug. At that time, the remaining 3 lamps had distinct amounts of oil.
Time passed by, and the room lost one more light while another mage died silently. At that time, the remaining 2 lamps did not only have distinct amounts of oil between each other, but also had distinct amounts from those 3 lamps at the time of the first mage's death.
Suddenly, the dark power removed the jugs from the last 2 mages, cutting the refill, and they continued their walk before one of them finally fell and was gone along with his light, leaving the lone survivor as the winner.
How many spoons of oil remained in the last alight lamp at the end?
The answer is 4.
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1 solution
After each round, 4 spoons of oil are burned up collectively, and 4 spoons are refilled collectively. This means that the total amount of oil in all lamps remains the same throughout, which is 4×2= 8 spoons
When the first mage dies, each of the remaining three lamps has a distinct amount of oil. Since one mage is dead, his lamp has no oil. Thus the remaining three lamps have 1,3,4 spoons of oil in some order.
When the second mage dies, the number of spoons in the remaining 2 filled lamps cannot be 1,3,4. However they must add up to 8 spoons. Thus there are 2 and 6 spoons of oil in the lamps.
The jugs are taken away and after two rounds, the lamp that only had two spoons left burns out. The other loses two spoons of oil. Thus the remaining lamp has 6-2=4 spoons left.