Sophisticated Timers
Alan has two identical timers, which both have a preset amount of seconds that tick down to 0.
These sophisticated timers have a weird property, when the number of seconds left on the timer is a multiple of 10, it makes a squeaking sound, and when the number of seconds left on the timer is a multiple of 30, it makes a barking sound.
Alan started both timers at the same time , and after both timers have reached 0, he makes the following statements:
(1) The second squeak can be heard 3 seconds after the first squeak .
(2) The last squeak can be heard 7 seconds after the second-to-last squeak .
(3) I heard a total number of 6 barks .
(4) At the time both timer started, there was a 21 second delay before the 2nd bark can be heard.
Let and be the initial preset amount of seconds on both timers. What is ?
Details and assumptions:
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As 30 is also a multiple of 10, when the number of seconds left on the timer is a multiple of 30, it will simultaneously squeak and bark
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is a multiple of 10 as well as a multiple of 30
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Alan has perfect hearing
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Alan has a perfect sense of time
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No actual dogs or mice are involved
The answer is 155.
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1 solution
We call the two timers TA and TB, based on their initial presets; and, WLOG, suppose A > B. From (2), we know that A = B + 7; this also means that the TA will bark at most one time more than TB (for TA to bark two more times than TB, we would need A > B + 30). But from (3) the total number off barks is even; so TA and TB must bark the same number of times, i.e. three times each. These barks must occur with 0, 30 and 60 seconds left on the timers, thus 60 < A, B < 90. Finally, since both timers bark the same number of times and A > B, the second bark must be A's, and it must happen with 60 seconds remaining; therefore from (4) it had to start with 60 + 21 = 81 seconds, and then B had to start with 81 - 7 = 74 seconds. So our answer is A + B = 81 + 74 = 1 5 5
It would seem (unless I missed something!) that clue (1) was unnecessary.