Math Circle: Splitting Squares

The Problem: Find all perfect squares less than 2000 that can be split into two parts (left and right), where both parts are also non-zero perfect squares.

Phase 1: The Setup

Instead of checking every number from 1 to 2000, we work smart. We categorize the numbers by how many digits they have.

Phase 2: The Hunt

Case A: 2-Digit Numbers

1x: 11, 14, 19 (None are squares)
4x: 41, 44...
check 49 (7²) → Split: 4 & 9.
Both 4 (2²) and 9 (3²) are squares.
✔ Match Found: 49
9x: 91, 94, 99 (None are squares)

Case B: 3-Digit Numbers (100–999)

Pattern 1: [Square] | [Square] (Right is 2-digits)

Left digit must be 1, 4, or 9.

1xx: Right part needs to be a 2-digit square (16, 25...).
Candidates: 116, 125, 136, 149...
Only 169 (13²) is a square. But split is 1 | 69. Is 69 a square? No.

Pattern 2: [Square] | [Square] (Left is 2-digits)

Left part must be a 2-digit square (16, 25, 36...).

16x: 161? 164? 169 (13²).
Split: 16 (4²) | 9 (3²).
✔ Match Found: 169
36x: 361 (19²).
Split: 36 (6²) | 1 (1²).
✔ Match Found: 361

Case C: 4-Digit Numbers (1000–1999)

Constraint: We are only looking under 2000. Therefore, the first digit must be 1.

Pattern 1: 1 | 3-digits

Check squares starting with 1... found 1225 (35²).
Split: 1 | 225. Is 225 a square? Yes (15²).
✔ Match Found: 1225

Pattern 2: 2-digits | 2-digits

Left part must be square starting with 1 → Only 16 works.
Found 1681 (41²).
Split: 16 | 81. Both squares.
✔ Match Found: 1681

Pattern 3: 3-digits | 1-digit

Left part is 3-digit square starting with 1 (100, 121, 144...).
Try 144... concatenated with 4 → 1444.
Is 1444 a square? Yes (38²).
✔ Match Found: 1444

Final Results

Number	Root	Split	Left Root	Right Root
49	7²	4, 9	2²	3²
169	13²	16, 9	4²	3²
361	19²	36, 1	6²	1²
1225	35²	1, 225	1²	15²
1444	38²	144, 4	12²	2²
1681	41²	16, 81	4²	9²

* Note: Solutions containing '00' or '0' (like 100 or 400) were excluded as "cheats".

Math Circle Challenge: The Splitting Squares