Kinesys lands on Staterion right next to the Market of Fair Time, where the stalls sell nothing but hourglasses: tall as streetlights, small as thimbles, filled with red sand, golden sand, even glowing sand. Clocks have been banned on this planet for centuries, so time is measured the Staterion way: grain by grain.
Dord is hungry, and when Dord is hungry he demands his Grox-root soup. “Nine minutes of simmering,” he declares. “Eight and it’s raw, ten and it’s glue. Nine. Exactly.” Liz lights the camping stove, M00N buzzes around the pot, and the soup starts to bubble.
The nearest merchant, though, will only lend them two hourglasses. “The big one measures seven minutes, the small one four. They flip in an instant, as many times as you like.” Liz suggests fiddling with the hourglasses for a while and starting the count whenever it suits her. Dord freezes her with a look: “The soup is already on the stove. The nine minutes start now, the moment you flip the first hourglass.”
The Riddle
You have two hourglasses: one measures exactly 7 minutes, the other exactly 4. Flipping them takes no time and you can do it as often as you like, but you have no other way to measure time.
How do you measure exactly 9 minutes, counting from the moment you flip the first hourglass?
Try it
Tap an hourglass to flip it: the stove lights up and the timer starts. Time only passes while you wait, until an hourglass runs out: that’s when you can flip them again. When you think exactly 9 minutes have passed, turn off the stove.
Hint
You don’t have to let an hourglass run all the way down: flip it partway through, and the sand that has already fallen goes back on top and measures the time that just passed all over again. Look at how much sand is in the 4-minute hourglass when the 7-minute one runs out.
Solution
Flip both hourglasses at the same moment, when the soup goes on the stove, and from then on act only when one of them runs out:
- Minute 4: the small hourglass runs out. Flip it right away.
- Minute 7: the big one runs out. The small one, flipped at minute 4, has let 3 minutes of sand through: 1 minute is left on top. Flip the big one.
- Minute 8: the small one runs out. The big one, flipped at minute 7, has let exactly 1 minute of sand through. Flip it again: that minute of sand goes back on top.
- Minute 9: the big one runs out. Exactly 9 minutes have passed.
In short, 4 + 3 + 1 + 1 = 9: the small hourglass measures the first 4 minutes and then, together with the big one, carves out a single minute, which the big one measures twice. Dord tastes the soup and, for the first time since anyone has known him, finds nothing to complain about.