Measuring with Jugs or What If Mathematicians Were Asked to Defuse Bombs
نویسندگان
چکیده
Script COMPUTER: I am a bomb, you have just armed me. SIMON GRUBER (over a cellular phone): I got you see the message. It has a proximity circuit, so please don't try to run. JOHN MCCLANE: Yeah I got it. We're not gonna run. How do we turn this thing off? SIMON: On the fountain, there should be two jugs. You see them, a ve gallon jug and a three gallon jug? Fill one of the jugs with exactly four gallons of water and place it on the scale and the timer will stop. You must be precise. One ounce more or less will result in detonation. If you are still alive in ve mins: : : JOHN: Wait: : : wait a second. I don't get it, you get it? ZEUS CARVER: No. JOHN: Get the jugs. Obviously we can't ll the three gallon jug with four gallons of water, right? ZEUS: Right. JOHN: All right. I know: : : Here we go. We ll the three gallon jug exactly to the top. Right? ZEUS: Uh huh. JOHN: Okay. Now. We pour that three gallons in the ve gallon jug, giving us exactly three gallons in the ve gallon jug, right? ZEUS: Right. JOHN: Now, take the three gallon jug, ll that a third of the way up: : : ZEUS: NO! He said precise. Exactly four gallons. As in any typical Hollywood action movie, John and Zeus (smart guys, indeed) have been eventually able to defuse Simon's bomb. But next time, it could be (die) harder! So it is high time for three wanna-be mathematicians to help our heroes to save their skin, and solve their problem in a far more general setting. Suppose we are given n initially empty jugs, each with a speciied positive integer capacity c i ; we can assume without loss of generality that c D .c 1 ; c 2 ; : : : ; c n / is an ordered vector (i.e., c 1 c 2 c n), xed from now onwards. We can perform three elementary operations (or steps) on the jugs, viz.:
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