Brave attempt Steve.
I did what Mrs R did, ie used combinations.
However, when I tried to explain this to Mrs D, she would have none of it. I was reduced to writing out all 36 four-digit arrangements of 0 1 9 and even now, she’s convinced there is some skull-duggary going on 
I’ll scan and print my list later today.
Well done Ravvie (and Mrs R)
Always seems counter-intuitive to me.
Only seems to work with 4-digit arrangements. Doesn’t work with 2, 3, 5, 6 etc, only four digits, apparently.
I look forward to seeing the list!
Counter-intuitive doesn’t even begin to describe it!
Hi Steve, I hope this helps. It was the only way that I was able to feel confident when I first encountered this teaser.
Ravvie’s contribution has also increased my confidence
I hope I haven’t made any mistakes in the above array.
My eyes seemed to loose focus towards the end
A bit like watching my football team ManU…I see it, but still don’t understand how it can be!
My thoughts exactly !
I think you are getting close to the definition of a ‘Brain Teaser’
This one is not counter-intuitive and doesn’t involve any tricks…
… Well, unless you don’t want to draw 99 Patterns and count all the squares
Going back to arithmetic progressions in A level, I have developed 3 formulae:-
The number of squares on the bottom row, where pattern number is “n”, is 1+2(n-1).
Taking the “bottom pyramid”, the squares in each row ascend from 1 up to 1+2(n-1) in increments of 2.
The “top pyramid” contains the same number of squares, less 1.
Using the formula for sum of terms in arithmetic progression, the formula for total squares for the 2 pyramids resolves as follows:-
n(2+2(n-1))-1.
Pattern 5 would contain 5(2+2(5-1))-1 which equates to 49, which can be seen to be correct.
So, pattern 10 would contain 10(2+2(10-1))-1 which equates to 199.
Pattern 99 would contain 99(2+2(99-1))-1 which equates to 19,601.
Well done Steve. 199 and 19,601 are spot-on !
Your formula above, produces the same results as mine, (which is 2n² - 1)
Thanks Don. I should have simplified my formula down to yours. I just felt I wanted to keep the “audit trail” nearer to the traditional formula for sum of arithmetic progression.
Unless of course you have a “neater” methodology.
Going back to the 4 digit codes problem, I still can’t get my head around it 
If we start with say the number 1234 there are 4! (or 24) separate combinations. If we then make the 4 a 3, ie the number 1233, we lose 12 repeated combinations. I can’t work out where the additional 24 come from? For the avoidance of doubt, I’m not doubting the correct answer - that is plain to see - I just need some help in understanding it.
With twice the number of squares in pattern N plus one square, you have enough squares to make a rectangle of 2N+1 times 2N-1 squares .
This gives Don result 1/2 ((2N+1)(2N-1)-1) = 2N*N-1
I know, I’ve had the same problem. I’m trying out a couple of possible explanations. I’ll post one of them if I can manage to describe things lucidly.
But so far, the best I have been able to do is write them out as I posted a couple of days ago !!
Well, I did approach it from a different direction. Vaguely recalled from school days more than 60 years ago.
Basically I looked at the successive differences, eg 6, 10, 14, 18, 22 …
Then at the successive differences again eg 4, 4, 4, 4 …
This suggests the progression involves n^2.
I 'll see if I can explain how I derived the co-efficient 2, thus deriving 2n^2 and the ‘-1’ and post it. I might have to dig out my old maths books to describe things clearly. At the moment, the co-efficient and the ‘-1’ just seemed to ‘fit-th-ebill’ by ‘observation’
It makes more sense if you look at it as two separate pyramids.
The bottom row of the bottom pyramid increments as follows 1,3,5,7,9, which is exactly the same as the increments between square numbers. The increment between the increments is a constant 2, which is the derivative of a square number.
You can see that the number of squares in the bottom pyramid is 1,4,9,16,25, which is the pattern number squared.
There are two identical pyramids with exception that the top pyramid has one fewer square (only the bottom pyramid contains 1 square; the top pyramid starts at 3).
So the bottom pyramid has n^2 squares. The top pyramid has n^2-1 squares.
In total there are 2n^2-1 squares.
You have shown correctly that 1233 has 12 combinations. The other 24 come from 1223 and 1123.
Thanks Ravvie, I do get it now! Still counter-intuitive though 
I hope the following description of my method helps. It is different to the way others have solved the ‘White Square Patterns’ teaser. And it reaches back to my school days.
PS Steve, I don’t think my method is neater than yours, its just different. It took more effort to describe my method than to implement it !!!
My daughter asked me to do a test run of a “connections” type quiz prepared for some of her colleagues. Took it to the pub last night for our regular Third Friday of the month meet. There’s a zoom meeting runs in parallel with this via a laptop on a table (a hangover from lockdown) so we effectively had two teams.
Here’s the first (of two). You have to connect four items and if possible identify the theme for that connection.
Both teams identified 4 connections fully, though not the same ones.
Willy.
Ring, It, Omen and Saw - all horror films?
