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Listener: 4164 by Kea

Posted by erwinch on 9 December 2011

I should think that many of us got a surprise to find Kea as the setter of a numerical puzzle – did we know that he was into numbers?  Another surprise was the sparseness of the clues, just one or two digits except for two clues with three.  The number and title (4164) was used as an example of how the puzzle worked, the clue for which would be 4.
41 × 4 = 164 
The two numbers 41 and 164 partially overlap (never completely) to give the solution 4164.
This was the first time that I had seen a puzzle of this form and I found it entertaining but once I became used to the idea it was very straightforward.  I prefer to use just a calculator, pencil and paper for these puzzles but admit that I did use a spreadsheet (MS Excel) as an aid this time. This was not quite a coffee break Listener but Excel was ideally suited here and it was not too long before the grid was complete:
The starting point was 3dn where the clue was 10 (henceforth the clued multiple is given in brackets).  The two numbers involved would have the form xx and xx0 combined to give the entry xxx0.  We later learned that x must be even to accommodate a multiple of two at 8ac.  So, 12ac (34) also ended in zero and only one fit was found: 15 × 34 = 510: 12ac = 1510
Now looking at 13dn (2) = 1??.  The first number ranges from 10 to 19 and the second from 20 to 38.  12 × 2 = 24 but 13 × 2 = 26 so 13dn = 124
25ac (150) ended with zero and the penultimate digit had to be even to fit with 19dn (2) so 25ac = ???00.  17dn (8) and 18dn (3) put similar constraints on the first and second digits respectively.  Again only one fit was found: 46 × 150 = 6900: 25ac = 46900 and 17dn (12) (= ???4) could only be 2264
Some three-digit entries such as 21dn (2) (= ??0) could be found by mental arithmetic.  A multiple of five was involved and it could only be 250
The higher the clued multiple the fewer the options were and 22ac (14) (= ???6) and 16dn (47) next caught my eye.  Two possibilities were found: {22ac, 16dn} = {3476, 1846} or {6966, 1799}
The highest multiple used was at 1ac (171) where five fits were found: 1ac = 11881, 12052, 23933, 35985 or 36156.  But we learned that the penultimate digit had to be even to fit with 3dn and 8ac: 1ac = 11881 or 35985, 3dn = 8880
And so it went with just a few hitches where I had to backtrack a little.  The final grid entry was 23ac = 2855
All that was required was a quick check to see that all entries were different (a general rule that seems to be rarely used) and that was it – short but sweet.
However, before I went back to my reading I had a look to see when the puzzle number would last have fitted here.  I had rather been expecting, hoping even, that No.4164 would be the 9th annual Elap numerical in a row but shall not think of it as the end of an era just yet since 4164 was specifically reserved for this puzzle’s number and title.  For convenience, I shall call these Kea numbers and going back twenty odd years, from No.3000, there have been 29 such numbers to date:
Of these only 3750 was also numerical: A Faulty Calculator by Aedites (29th November 2003) almost exactly eight years ago.  The next Kea number is 4210, which is due to be assigned to a puzzle published on the 6th October 2012.  So this puzzle was presumably set some time after 2003 and indeed Viking may have had a hand in it.  It was just waiting for the right number to come along.
Here are the 30 Kea numbers that follow 4210:
I found a Website that told you how many Saturdays there are in any year up to 2038 and cross checked these with calendars up to 2040.  We average 52 puzzles a year.  There is no Listener if Christmas Day is a Saturday, which will cancel out the 365th day of the year, but Leap years will give us an extra puzzle every 28 years.  The 30 numbers above take us up to November 2040 and only one is due to be numerical – 4998 on 20th November 2027.  Starting in February this year the numerical puzzles were moved from the final to the penultimate Saturday in February, May, August and November.  The reason given was that they wanted to avoid the numerical puzzles falling on Bank Holiday weekends but this seemed unnecessary given their wide popularity.  I wonder, could the real reason be that they wanted to fit this puzzle in?  Under the old scheme, 4387 (27th February 2016) would have been numerical but perhaps that was considered too long to wait.   I am just glad to have them at all and do not really see the need for such a rigid timetable.  The only possible reason for it that I can think of is to placate the vocal minority who see no place for numbers in the Listener and would rather have them permanently removed from the calendar – they have no need to buy the paper on these dates.

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