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Hi, Dryden, I think we know by now from previous replies that all the grid entries are primes of the form 4n+1. The mathematician Pierre de Fermat showed that any odd prime p can be expressed as the sum of two squares of positive integers if and only if p = 4n+1.

For the endgame, I checked whether any of the 20 9-digit numbers in the grid was divisible by 3 by calculating the digit sum of each of the numbers. If the digit sum is divisible by 3, then so is the number itself. In all of the symmetric pairs of 9-digit numbers, at least one of the pair was divisible by 3, and hence not a prime, except for the two numbers on the diagonals, which both had to be read downwards to satisfy the common feature.

Then came the slog. I took the smaller diagonal number and worked out the integer closest to the square root of half the number. I then took every 5-digit number from 10,000 to the integer I’d just calculated, and set up an Excel spreadsheet to calculate the square root of (the 9-digit number less the square of each 5-digit number) in my range, then inspected the results to look for the only answer that was itself an integer. Once I’d found the two 5-digit numbers in the grid, it was clear what the other pair of 5-digit numbers was, and I checked that the sum of their squares gave the 9-digit number on the other diagonal.

I’d been using a website for factorisation and checking whether numbers were primes, but the sum of squares website referred to in earlier replies would have saved a lot of tedious work on Excel. It will be interesting to see when the solution comes out if there was a more straightforward way of doing this puzzle than a lot of Excel calculations.

A fine numerical that could be solved in the comfort of an armchair with only a standard scientific calculator. Lots of nice deductions and reasoning cut out the need for any computer use. Thanks Arden.

Thanks to you both for your descriptions of how you went about it. I suspect if I hadn't come to this site and learned about that internet site I would still be calculating. I did originally search for an appropriate site but ll I got was a lot of sites that didn't quite offer what was needed.

I have a nifty app on my cell phone that I use a lot on numerical puzzles like this, called "Prime Factors" by tappopotamus. It's elegant and easy to use and very fast; it will instantly show you if a number up to eleven digits is prime, and give you the prime factors if it's not. It helped me find the numbers to highlight pretty quickly.

The app is particularly handy when the numbers you need to check are all within a range of a few hundred, as you can quickly scroll up or down through the numbers rather than type in each number you need to check. Prime numbers are marked by big circussy red dots, which makes it easy to quickly find all the primes within a certain range or the nearest prime to a given number or something like that.

I also made a lot of use of a spreadsheet at first, to run through all the possibilities quickly and let me search for the valid possibilities. After a while I wrote a short program in BASIC that did the same thing but faster. Whether it actually saved me more time than it took me to write it and get the bugs out of it, I'm not quite sure.

I am interested in how people solve these numerical puzzles. I tend to resort to MS Excel due to the large volume of calculations that is typically required. How many people on this forum use:

1. Pencil & paper only
2. Calculator
3. MS Excel
4. Helpful websites
5. Other (abacus?)

pencil paper, calculator , lists from web of primes etc. I am not clever enough to write programs to solve but some people on here are!

Often excellent links which help are given on this thread .....very useful ones this week!

Hi Kitsune.
On Saturday evening I solved the first couple (7D, 6A) mentally while driving home. I wasn't going to achieve 9A mentally, so the remainder of my journey was spent on planning how to identify the clues which are the products of integers, and methodology. At home I used pen & paper, calculator, online factor finding and tables of squares, primes. There's no mystery to BASIC programming, it's one of the easiest programming languages; you could teach yourself to use it in just a few minutes. But I chose instead to use spreadsheet. Programming logic is little different from using spreadsheet formula. And errors can be introduced into each by slips such as specifying the wrong column of data (in a program that would equate to a table of data) !

I usually solve on my laptop during my long commute by train, using Apple’s word processing/page layout program called Pages. This is a little more trouble than pencil and paper, but has some advantages as well. On one page I place the PDF of the puzzle (downloaded from the website) and mark it up as I go; on the other I make notes and keep track of my train of logic. Then my puzzles are always all in one place, whether they’re in progress or completed, and when I’m done I don’t have to transcribe my solution onto a clean grid; I just print my filled grid and mail it. Because of my commute, I often have to solve in many 30-minute bursts rather than in one or two sittings, and I’m very absent-minded (and not getting any less so as I age), so I’ve learned that with a numerical I need to make good notes as I go so that I can pick up my train of thought; it’s also very helpful if I reach an impossibility and need to go back and find my mistake in logic. Another advantage: No limit on how many times I can erase before the paper wears through.

I keep a lot of lists of numbers in my notes app on my phone — primes, squares, cubes, Fibonacci numbers, and so on. Basically, any time I need to search the internet to find a list of numbers, or I generate a list of numbers in a spreadsheet, then if I think there’s a chance it will be useful again on some future numerical, I copy it into my notes so I can find it quickly next time.

I usually use just a calculator, but if it becomes clear that I’m going to need to do a lot of similar calculations or brute force my way through a great many possibilities, I’ll open up a spreadsheet program or in extreme cases I’ll open up a BASIC emulator and write a short program. But it’s more fun if I can find a path that doesn’t require those things.

Kitsune

Hi. I’m a bit of a maths geek so always try to get as far as I can with just pen and paper. Filled the grid and worked out the diagonals must be the primes just using the brain, no calculator or spreadsheet or internet help. For that reason I thought this was pretty good - normally that is impossible.

Then used the internet to check the two diagonals were in fact prime, then a spreadsheet to find the two sets of smaller numbers. Feel like I should have guessed that is where they would be having done this, so could have done this last step unaided as well, which would have been very satisfying.