(Part 1 has some background to the problem)
Squishy squares are squares like 81, 2025, 9801 whose square root is the sum of the top half digits and bottom half digits.
So \(\sqrt{81} = 9 = 8 + 1\)
We can do some analytical tinkering on the problem as follows:
Let \(x, y\) be the top and bottom half digits of the square, so the squishy square takes the form \(10^nx + y\) and \(z\) is the root.
\[z = x + y \\ z^2 = 10^nx + y \\ (x + y)^2 = 10^nx + y\]Where
\[0 \leq y \lt 10^n \\ 10^{n-1} \leq x \lt 10^n\]We can reorganize this as a quadratic in x:
\[x^2 + (2y - 10^n)x + y^2 - y = 0\]Using the quadratic formula we can write
\[x = \frac{(10^n-2y) \pm \sqrt{10^{2n} - 4*10^ny + 4y}}{2}\]For y = 0 we get \(x = 0\) or \(x = 10^n\), which aren’t valid
For y = 1 we get \(x = 10^n - 2\) which explains 81 and 9801 and so on.
Sadly, I can’t think of any further to go with absolute values of y.
I’ll try something else later …
I was revisiting my older article and realized that I had missed an opportunity to find more patterns in the numbers by not printing the roots alongside the squares.
After a quick refactor of the code I printed out the roots and squares (see end of article).
I was about to disappear into several rabbit holes (Look at 45, 4950, 351352, 499500, there’s definitely something there) when I started to do my analytical tinkering.
9 -> 81
45 -> 2025
55 -> 3025
99 -> 9801
703 -> 494209
999 -> 998001
4950 -> 24502500
5050 -> 25502500
7272 -> 52881984
7777 -> 60481729
9999 -> 99980001
77778 -> 6049417284
82656 -> 6832014336
95121 -> 9048004641
99999 -> 9999800001
318682 -> 101558217124
329967 -> 108878221089
351352 -> 123448227904
356643 -> 127194229449
390313 -> 152344237969
461539 -> 213018248521
466830 -> 217930248900
499500 -> 249500250000
500500 -> 250500250000
533170 -> 284270248900
538461 -> 289940248521
609687 -> 371718237969
643357 -> 413908229449
648648 -> 420744227904
670033 -> 448944221089
681318 -> 464194217124
791505 -> 626480165025
812890 -> 660790152100
818181 -> 669420148761
851851 -> 725650126201
857143 -> 734694122449
961038 -> 923594037444
994708 -> 989444005264
999999 -> 999998000001
4444444 -> 19753082469136
4927941 -> 24284602499481
5072059 -> 25725782499481
5555556 -> 30864202469136
9372385 -> 87841600588225
9999999 -> 99999980000001
36363636 -> 1322314023140496
38883889 -> 1511956823764321
44363341 -> 1968106024682281
44525548 -> 1982524424700304
49995000 -> 2499500025000000
50005000 -> 2500500025000000
55474452 -> 3077414824700304
55636659 -> 3095437824682281
61116111 -> 3735179023764321
63636364 -> 4049586823140496
69115816 -> 4776996021345856
74747475 -> 5587185018875625
75247525 -> 5662190018625625
80226927 -> 6436359815863329
80726977 -> 6516844815558529
83409436 -> 6957134013838096
86358636 -> 7457814011780496
88888888 -> 7901234409876544
91838088 -> 8434234407495744
94520547 -> 8934133805179209
99999999 -> 9999999800000001
332999667 -> 110888778222110889
432432432 -> 186997808245434624
567567568 -> 322132944245434624
667000333 -> 444889444222110889
765432099 -> 585886298179545801
999999999 -> 999999998000000001