Brain-Teaser 658: Fiendish device
From The Sunday Times, 17th February 1974 [link] [link]
“Moriarty speaking”, said the voice on the telephone to the Prime Minister. “As you have rejected my demands, a hidden bomb with destroy London. I’m particularly pleased with the detonating device”, he went on, chuckling fiendishly, “it’s designed to give me time to get away before the explosion. There are 60 switches (all turned OFF at the moment) arranged in a ring so that No. 60 is next to No. 1. Whenever any switch changes from ON to OFF it causes the following switch to change over virtually instantaneously (from OFF to ON or vice-versa). As soon as I put down this phone I’ll activate the device. This will automatically put switch No. 1 to ON, then one minute later to OFF, then one minute later still to ON, carrying on in this way after each minute changing switch No. 1 over. As soon as every switch has remained in the OFF position for 10 seconds simultaneously the bomb explodes. So goodbye now — for ever!”
The Prime Minister turned anxiously to Professor D. Fuse who had been listening in. “When will the activating device set off the bomb?” he asked.
What was the Professor’s reply?
This puzzle is included in the book The Sunday Times Book of Brain-Teasers: Book 1 (1980). The puzzle text above is taken from the book.
[teaser658]
S2T2 
Jim Randell 9:29 am on 5 January 2021 Permalink |
If there were only 3 switches, this what would happen:
If we read the switches backwards we see the system of switches operates as a binary counter starting at 001 (= 1) and counting up to 111 (= 7) after 6 minutes.
After 7 minutes the counter would reach 8 = (000 + overflow), but the switches are arranged in a circle, so the overflow bit feeds into the least significant bit of the counter, so state 8 corresponds to state 1 (= 001) and we are back where we started.
The counter cycles through the non-zero states 1-7 every 7 minutes, so never achieves state 0 and the bomb will never go off.
So, with 60 switches we have a 60-bit counter which cycles through the states 1 – (2^60 − 1) every 1152921504606846975 minutes (= 2.2e+12 years), without ever reaching zero.
Solution: The Professor’s reply is: “Never”.
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