If in doubt, cheat become resourceful

Welcome back to George vs the Listener Numberword. Four times a year there’s a little numerical challenge, and back in “the day” I really looked forward to them, because it was typically all logic and I’m fairly good at that.  Sometimes a twist is added and then I tend to do a little worse.  This year  I did well on the first one, which did have a nifty trick to it.  However if there’s one name that strikes fear into me, it’s seeing that the setter of the puzzle is Oyler.

In the year before I started keeping track, the only numerical I missed was 3944: Dedication, that required a cipher as well as the number solution.  Then came 3984:  Odd One Out, which gave me a rare opportunity to come up with not just one, but two completely empty grids!  Oyler did an about-face with 4023: Pentominos where I solved the entire thing in one session without even a calculator.

So Which Oyler showed up this time?  First off, I thought there was some sort of misprint.  I couldn’t see anything indicating which are across, which are down, and what was going on.  I’ll confess, when I first got it out to work on it, I decided to give it a miss and keep bashing away at Auctor’s puzzle from the week before.  Not sure if that was a winning move or not.

Another careful read of the preamble… each row looks like it matches one of these funny relationships where x squared plus y squared gives a string of digits. And there’s no zeros anywhere – maybe Oyler read my blog about how all the zeros in Pentominos made a few lines a giveaway and decided that I deserved no zeros!

Wow… how to get started on this… well squaring 5 digit numbers is more like to give a 10-digit number than a 9-digit number, so let’s see how high these 5 digit entries can go… looks like 31622 (actually lower because we have to add in the square of the 9-digit numbers) is our upper limit.  It also makes 152,000,000 or so the lower limit for the 9-digit sequence but that doesn’t seem as useful.

Let’s find those squares, you know how the way to a numerical puzzle’s heart is through its squares or cubes or something of the sort.  The squares are associated with P & J… J looks like a nice starting point – J is a two-digit number where 6J squared and 7J squared have to be three digit sequences with no zeros or repeated digits… irrelevant the only one that gives a 7J squared less than 1000 is 11.  Woohoo! J = 11!

Next thing that struck me was that a few entries are multiplied by 5.  So not only do they have to be odd numbers, but they have to end in 5.  There were a lot of random 5’s scattered through the X, Y, Z part of my page.  That brings up a chance to find h.  h has to be odd, 3 digits, no repeating digits in h, 5h or 6h (all of which have to be under 1000).  A bit of spreadsheeting throws up 127 as the only possibility for h, and so the first row is 635,127,???

The last three question marks are m’, and the first digit of m intersects with G – 5G can’t cotain 7,2 or 6 and the first digit of G is 2 (from h), so that limits G to 29.  So the last of my three question marks is a 9.

Back to excel… put in both 625127489 and 625127849.  OK… now I’m adding squares here, so to add to 9, it has to be 4+5 (squaring numbers ending in 2 or 8 and 5).  The max value for the four digit entry squared is 97535326, so subtracting that from 625127849 and taking the square root means that the minimum value for the five-digit entry is 21971 and the max is 25201.  So we can throw out numbers ending in 2.

See, said I was pretty good at logic!  So now excel is set up where I enter in my five digit number 2???5 or 2???8 (no repeat numbers, no zeros).  The magic of the spreadsheet squares my number, subtracts it from both 625127489 and 625127849 and takes the square root.  Hopefully sooner rather than later that square root column will show a whole number.

VOILA!!!! After a little bit of keybashing, the first whole number that appeared was if I typed in 23475.  Not only that, but the whole number was 9168 and that fits perfectly.  A good logician would have gone back to check, but I’m thinking there can’t be too many of this bizarre combination, so in it goes.  Even better news is that 23475 is divisible by 3, to give 7825, and the prime factors of 7825 are 313, 5 and 5, so that takes care of s (313) and f (25).

Bloody hell that was a lot of work for one number!

And it doesn’t look like it’s going to get any easier… same spreadsheet looks like it’s going to work for the fifth row, where I know 726,???,145.  Expand it to the 6 possibilities for those three middle digits and start at 23958 for the 5-digit number.  This one took a lot longer, but eventually I got to 25413.  Oh great… that doesn’t resolve A or r and getting another row of numbers doesn’t put a SINGLE digit in the grid.  This is getting frustrating.

I worked a little bit on the three-digit entries, sorting out that p had to be 157 or 159, P looked like it had to be 13, but I was still getting nowhere. I had to sort out another row.  I had ??5,???,721 as the number for line 3 and knew the 4 digit number was a multiple of 48 and the 5 digit a multiple of 25 (and at least 313*25).  Well that’s not a great start.  G’ is 92, W is at least 12 so the first three digit sequence has to be 3?5.  So in go 12 possibilities, and this time I tried working up multiples of 87 to find the 4-digit number first.  This one took even longer, but there it is… 2436 (making n=28) makes the 4-digit answer 18975 (thankfully I have enough digits in to reduce that down to M = 385 and z = 61) and the 9 digit string is 365(so W = 19),984(doesn’t help me on d or b),721.

Still with me? Row four next… the 4-digit number is a multiple of 145, and the 5-digit number is 9*(3?6?).  The string is 825,???,314 (p being settled by knowing 2p cannot have an 8 in it).  Working with those multiples of 145 eventually I hit on 3915 and a string of 825,697,314

I’d like to say that I persevered like an Excel demon with fingers of steel through the rest of the solving process, but at this point I had a change in fortune.  I’d tried googling all sorts of variations on “9-digit pangrams”, “squares that add to nine-digit sequences”, “excel is killing me get me out of this crossstring”, with no luck.  So then I turned to the 9-digit sequences.  I got a few Romanian phone numbers, one lead that I think someone was using an online Mathematica sort of thing on 635127849, but it wasn’t until I put 825697314 into Google that Deus ex Interweeb appeared in the form of this site.

A handy-dandy table showing that there’s 72 options!  Quick copy and paste into excel and now I’ve got the 9-digit strings that fit.  The other seven sets were now easy to find and a little mathematics to work out the grid entries, and you can take the high road, but I’ve invested hours in random Excel bashing and let’s just call this completed and move on, eh?

my "working" for Listener 4088: Digimix by Oyler

Some of this was painstakingly put together

Wow, Oyler.  That was a DOOZY! And my fingers would have fallen off if I hadn’t found that site eventually, but I think I’ve shown that I could have gotten there in the end. I wonder if there’s an easier way I didn’t know?  If I don’t have to spreadsheet for three months I’ll be happy.  What’s that? I have to do the budget for next year?  Bugger!

My final grid for Listener 4088: Digimix by Oyler

My final grid for Listener 4088: Digimix by Oyler

Victory to George and the blip from last week may not be the start of a slide after all.  2010 tally:  George 19, Listener 3.  Current streak:  George 1.

Entering “Digimix” into YouTube brings up this pretty dire-sounding New York Latino Electronica thing.  How much can you handle?

Feel free to leave comments below, ca va dernier semaine pour “What a difference” avec Kevin.

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10 Responses

  1. And to leave insult to injury, looks like I can’t add 10 numbers together properly. Sorry, Oyler

  2. Hi George,
    I too struggled through this with the aid of multiple spreadsheets. I don’t know if there are any elegant shortcuts but by the time you’re invested in Excel it is just too tempting to use it to mine through possibilities rather than applying rigorous (and in my case potentially error-prone) logic.
    When I was all done I fell to wondering just how many possible values there are for P, Q and X_Y_Z. So I wrote a quick Java program to find them and then re-solved the puzzle using the generated list. That way took me about twenty minutes.
    I do sometimes wonder if these puzzles are set up on the assumption that solvers will use such computational aids.

    …Robert

  3. I solved it with a perl script from the outset, and completed the grid correctly but my total at the bottom was 72294, 900 out. I’ve double-checked my figures and still can’t see where I went wrong. I only had 8 numbers though, not 10.

  4. It is interesting that my pseudonym strikes fear into you. Solvers of The Magpie puzzles reckon that my pseudonym gives them a source of comfort!! Oyler

    • Hi Oyler! Thanks for checking in – I’ve not tried any Magpie numbermericals (or other Oyler puzzles that appear there), I’m just saying that of the regular numerical setters, yours are the ones I have the worst track record on. I would also like to award myself something, I think the last sentence could be the most supreme butchering of the English language ever.

  5. Hi Guys. I used this: http://codepad.org/DuNNQYrh.

    I went a bit off track by having h dn as 177 although I had the other part M dn as 313. Most of what I had was correct in the end. Pandigitals 1-9 took into Excel and had a formulae of 78 values for P and Q – break them up into what’s needed and it all became easy -Once you know whats going on!

  6. […] from very easy to very hard.  A walk through previous Oyler Listeners shows that last year’s Digimix was very hard, and I had to resort to on-line aids to get through it. Oyler checked in on that one, […]

  7. […] Elementary Number Theory, where a bad slip near the end made my grid incorrect.  Before that was Digimix, where I was almost right, but couldn’t add up some long numbers so missed the final step.   […]

  8. […] 2X2X2 (which I got), before that was Elementary Number Theory, which I messed up near the end, Digimix, which I messed up near the middle, Pentomino Factory, which I got, and the less said about Odd One […]

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