Enigma 272: Squares and crosses
11 April 2015
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From New Scientist #1419, 30th August 1984 [link]
The Returning Officer was a terrible tease. After the votes had been counted he addressed the three candidates as follows: “By a strange coincidence each of you has polled a number of votes which is a perfect square. I have told each candidate separately how many votes he received — you all got some I am pleased to say. The winner got exactly 50 per cent of the votes cast. As you know the total electorate is 1000”. He then asked each candidate in turn if he could deduce the full result.
The Amity candidate said: “I cannot even deduce the percentage turnout”.
The Brotherhood candidate then said: “I know how many votes were cast for each candidate”.
The Comradeship candidate said: “So do I”.
The Amity candidate then said: “So do I now”.
The three candidates were of course perfect logicians and entirely honest.
How many votes did each receive?