Brain-Teaser 889: Counting the hours

From The Sunday Times, 20th August 1978 [link]

At school the other day, little Johnny was working with one of those boards with twelve clock-points regularly spaced round a circle against which should be put twelve counters showing the hours.

He was, in fact, in a bit of a daze about the whole thing and the only hour he was absolutely dead sure of was 12 whose counter he correctly placed. As for the eleven others, if the truth be told, he just put them around at random.

But Jill, his teacher, spotted some curious things. She first noticed that, however she chose a quartet of counters which formed the corners of a square, their sum was always the same.

Next, she saw that if she formed numbers by multiplying together the counters at the corners of each square, one of those numbers was more than six times one of the others.

She also observed that the counters in each quartet were, starting at the lowest, in ascending order of magnitude in the clockwise direction, and that 12 was not the only correct counter.

In break, she reported all this to her colleague, Mary, adding “If I were to tell you, in addition, how many hours apart from the 12 Johnny had got right, you could tell me which they were”.

Which were they?

This puzzle was included in the book The Sunday Times Book of Brain-Teasers: Book 1 (1980, edited by Victor Bryant and Ronald Postill). The puzzle text above is taken from the book.

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