This week’s puzzle is split into two parts: the first is an old puzzle that many might not have seen, and is worth a look in its own right; however the second part is new, and is slightly trickier.
The problem concerns a bath. With the plug in and the cold tap turned on full the bath will fill in three minutes. The hot tap is slightly slower, and with the plug in and the tap on full the bath will fill in four minutes. With the taps off a full bath will empty in twelve minutes when the plug is removed.
So how long will it take to fill the bath with both taps on full and the plug removed?
Assuming that you have solved this part, the trickier subsidiary is . . .
If the bath starts half full how long will it take to fill completely if both taps are turned on full and the plug is removed?
To solve the first puzzle is to consider what happens in one minute. With the cold tap on full for one minute the bath will be one third full (as it takes three minutes to fill completely). By the same logic, with the hot tap on full for one minute the bath will be a quarter full. And in one minute one twelfth of the bath will go down the plug hole with the plug removed.
With both taps of them going full blast and the plug out, after one minute there will be 1 / 3 + 1 / 4 – 1 / 12 of the bath full. Which is (4 + 3 – 1) / 12 = 6 / 12 or half full. Thus it will take two minutes to fill the bath.
Where it becomes trickier is the second part. The pressure in a liquid is proportional to its depth. So our feeling is that the bottom half of the bath will empty more slowly than the top half. Thus the top half will take more than a minute to fill. But how much more we're not sure.
However many contributors feel that the difference will be negligible and one minute is the answer.
Like we said, not an obvious puzzle at all!