**From New Scientist #1358, 19th May 1983** [link]

I displayed a three-figure number on my calculator. My son looked at the calculator upside-down and said:

“I can see a three-figure number too, and it’s less than yours”.

I added 12 to my number.

“I can still see a number, again less than yours”, he said.

I multiplied my latest number by 6.

“More this time”, he said.

Finally I added 11.

“LESS”, he said, much to my surprise, and then we both laughed.

What number did I originally display?

[enigma212]

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My first thought was that LESS, looks suspiciously like 5537 displayed upside down on a 7-segment calculator display.

If this were the case then, working backwards through the problem we get:

5537, displays as “LESS” upside down.

Subtracting 11, gives:

5526, displays as 9255 upside down, and 9255 > 5526.

Dividing by 6, gives:

921, displays as 126 upside down, and 126 < 921.

Subtracting 12, gives:

909, displays as 606 upside down, and 606 < 909.

Solution:The first displayed number was 909 (the right way up).This Python program looks for sequences that satisfy the conditions up to the final “LESS” statement, and shows there are only three candidate sequences, none of which end up with a number that can be successfully read upside down to give another number. It runs in 34ms.

I recently heard an edition of “Puzzle Panel” on BBC Radio 4 Extra (originally broadcast in July 1998) where Victor Bryant (the real Susan Denham) gave a similar puzzle:

The answer, of course, being: 5537 × 3 – 30 – 13 – 3 – 3 – 3 – 3 = 16556.

And we can verify that the first 6 inverted numbers are indeed greater than their correctly oriented counterparts.

95591 > 16556, [+3] 65591 > 16559, [+3] 29591 > 16562, [+3] 59591 > 16565, [+3] 89591 > 16568, [+13] 18591 > 16581, [+30] [÷3] 5537.

Victor commented that in the past he had set a puzzle based on this idea and received a letter informing him that “there is no place for trickery in puzzles”.

Do you know why he chose to use the pen name “Susan Denham?”

I think it’s because “Sue Denham” sounds like “pseudonym”.

See

Enigma 1377.If you’re in the UK and you want to hear the episode I’m talking about it’s available for the next 13 hours at [ http://www.bbc.co.uk/programmes/b00yjr5c ]. The puzzle above starts about 8 minutes in.

Without this tip on the trick involved I could only show that numerically there are no solutions given that the only digits I can invert are 0,1, 2, 5, 6, 8, 9.

Congratulations on spotting LESS as a “first thought”. It was very much a last thought for me, when visually inspecting the few possibilities which fell at the final hurdle.