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Listener 4712: Sequences by Elap

Posted by vaganslistener on 11 Jun 2022

Here beginneth the struggle of a mildly number-dyslexic arts graduate with the form of torture known as a Numerical… Elap is a master of the craft and let not my woes detract in any way from a superbly constructed puzzle. I can’t begin to imagine how anyone can construct these things. But it has to be said that the preamble alone was enough to fill me with dread (if also the excuse to open a decent bottle of something). And if I am allowed a slight grunt it is that discovering a sequence with unknown rules and parameters and one which is not an old chestnut like Fibonacci is not at all trivial for the non-mathematician. I gather those that code can devise programs to assist with these things. I run to basic Excel which is quite a help – but also still quite a grunt, with no real way of being sure you are on the right lines until it all falls together at the end.  But hey-ho and off we go. I’ll tell you the story how (I think) I cracked it, which is probably not all the pure maths route intended…

I started of course by Googling a list of what I discovered are called “power primes” which came up easily and at least established the set of values for the letters. Thinking about the rule, both the wording and the limitations of what could be said in the message led me to believe that it would have to be some simple operations on the digits of the numbers: add, multiply, square or whatever. The letters of the message rules out ADD, SUBTRACT and SQUARE but not SUM and POWER (which was of course staring us in the face as 2a). But all of that would come later: where to start with finding some values?

The clues laid out as equations obviously gave additional information so I started with those. 25d and 26d gave Y=3, S=4, I = 16: a good start! 26a gave 3 values for O: only 49 worked; U & M are 5 & 9 or 9 & 5. Making progress!

Time to remember to look at that emerging message: I decided that swapping between that and the maths would be my Secret Weapon.           So far I can see it beginning –YSU–M…. : it looks like SUM is going to be in there, and of –Y patterns TRY looks likely.

Back to the clues. 27d confirmed E=83, C=107, T=1. Then the ones with SUMS in (8a, 17a, 26a, 1d, 3d) and 31a followed easily. Now the message is TR/rY SU.mM..I.N …….O………E…..C………………… TRY SUMMING POWER? What does that mean? Something to do with each digit raised to a power. Is the power the parameter that changes? 

But I can’t do anything with that yet, so I decided to make a humungous Excel table of numbers to 20K with each digit raised to power of 2,3,4,5 and summed. It wouldn’t catch the biggest numbers but it would in due course allow me to check what number may follow another in sequence and search for where any given number cropped up. The split 5-digit numbers would be a real googly, but hopefully the smaller ones could still be found.

That was a welcome diversion from the clues, because I was not too sure where to go next. 10a gave me a, then 4d gave G (which must divide by 9) then 21d with r=2, and 5d as 514 for 5d. 

13d gave N+352 = x7x; the message meant N was 17, 19, 23, 25 with the last three fitting the x7x pattern to give 371/375/377, so N = 19, 23 or 25            

5a could now be simplified to 87 + 8e with pattern 5xx and e =59, 61 or 64. 2d was therefore = 786 + 59, 61 or 64 = 845, 847, 850: 64/850 is excluded by pattern; so e = 59 or 61  

22a = 652 + e with pattern x 1/5/7 x so was 711 or 713; 22d was 701 = -79 + 16R – 4e; 16R-4e=780; only e=61 gives an integer for R; so e = 61; R = 64     . Then I could fill in 5a = 575; 22a = 713; 5d = 514; 2d = 845; 14d=493     

Now 14d (493) is the start of a sequence. The message is now TrY SUmMING O e R EaC with APWoFH unused. So I plumped for TRY SUMMING A POWER OF EACH [DIGIT]. Time to test the theory. The fourth power sequence generated by 493 proved to be 6898, 16049, 8114, 4354, 1218, 4114, 514, 882, 8208. 8208= 31a and 514 = 5d. Things were looking up!

I noted that of the unallocated letters A and P were in the range 29 > 47, W was 53 or 59, o and F were in the range 67 > 81, and H in the range 109>199.

13a = 344 so P + H = 216 and must be 47 and 169, 43 and 173 or 37 and 179. The 7xx pattern of 6d excluded 37 & 179; 47 &169 gave 737; 43&163 gave 721. 11a = P + 87 = 134 or 130; so 6d must be 737; 11a must be 134; 5d was 54; P was 47; H was 169. 7d now gave F+W=134; W=53, F=81. 15d=882. 18a and 20a meant A = 43, and 9d gave o = 79. All the letters were now found and grid could be completed.

(I’ll be staggered by the way if I’ve recorded that process fully and correctly – life is to short and the dyslexia too dangerous – but it gives an idea of the journey.)

Now the other two sequences needes to found. The three were likely to be based on ^2 ^3 ^4 or ^3 ^4 ^5 if the puzzle was actually gone to be solvable and the number lenghts kept under control. ^2 hadn’t looked promising when testing the first sequence so I decided to try the second option and start with ^3. 

I expected mostly 3-digit numbers so started to work through my Excel chart. The first on the  list of cubes that appeared in grid was 155, which generates 251, 134, 92, 737, 713, and 371which (Alleluia!) all appeared in the grid. At that point I nearly became unstuck, because I didn’t take the ‘at least seven numbers’ of the preamble to heart and check for predecessors to 155 – but there they were, 845, 701 and 344. Whoops!

I then had to find a way of discovering the ^5 series which was always going to be the trickiest. I decided to try the 3 digit numbers for sequences, starting at smallest. No joy from 169 or 23, but 472 generated 17863, 57595, 85231, 36169, 74845 and 54748, and to my amazement I was home and dry.12a 155 and 23a 472 needed to be highlighted, and the last numbers were 371, 8208 and 54748.

Serendipity and luck played a major role in that. My guesses could easily have been wromg. I wonder if there was a more secure way home – and kudos to those who managed to find it – and thanks again to Elap. Rather a snaky route for me but it was good to make the journey and get home safe at the end.            

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