Clock times using 6 numbers

A number puzzle has recently been given by Column8 in the Sydney Morning Herald newspaper.

The proposer wondered what “clock-times” between 1:00 am and 8:00 am he could make with the six numbers 75, 25, 1, 3, 5 and 7 (reminiscent of the puzzles on the TV game show Letters and Numbers).

For example, 7:40 can obviously be calculated as (75 – 1) • (3 + 7) i.e. 74 • 10 giving 740 (7:40 on the clock, so to speak). This also means that numbers like 570 or 799 do not count since they are not clock times.

It was actually claimed that every time except one had at least one solution. What a laborious thing to verify, without a decent little computer program of course.

However, starting from the top and working backwards, I soon found several gaps:

8:00 (75 + 25) • (1 + 7)

7:59

7:58 (75 – 25) • 3 • 5 + 7 + 1 OR (75 – 25 + 1) • 3 • 5 – 7

7:57 (75 + 25 + 3 + 5) • 7 + 1

7:56 75 • (3 + 7) + 5 + 1

7:55 75 • (3 + 7) + 5

7:54 75 • (3 + 7) + 5 – 1

7:53 ((75 – 25) • 5 + 1) • 3

7:52

7:51: 75 • (3 + 7) + 1

7:50 75 • (3 + 7)

7:49 75 • (3 + 7) – 1

7:48

7:47

7:46 75 • (3 + 7) – 5 + 1

7:45 75 • (3 + 7) – 5

7:44 75 • (3 + 7) – 5 – 1

7:43 (75 – 25) • 3 • 5 – 7

7:42 (75 + 25 + 1 + 5) • 7

7:41

7:40 (75 – 1) • (3 + 7)

The proposer has now revealed that all of the numbers except 7:59 have solutions. I was amazed.

After more scribbling, here’s my solution from “outside the square” for this number:

7:59 (75 + 1) • (3 + 7) – ( √ 25 ) / 5 = 76 • 10 – 1

Who can deny that square root is an arithmetic operation?

There remain the other incomplete entries – any solutions?

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2 thoughts on “Clock times using 6 numbers

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