1. Other Ways of Making 747
A 3-digit number abc added to its reverse cba gives:
(100a + 10b + c) + (100c + 10b + a) = 101(a + c) + 20b.
For no carrying, a + c = 7 and 2b = 4 (so b = 2).
- Given: 621 + 126 = 747
- Second way: 522 + 225 = 747
- Third way: 423 + 324 = 747
2. All Totals Between 700 and 800 (10 Totals)
Any sum between 700 and 800 must have hundreds digit 7. This can happen in two ways:
- Case A (No carry to hundreds): a + c = 7.
The tens sum is 2b. Since b ranges from 0 to 4 (so no carry):
• b = 0 → 707 (e.g. 700 + 007, or non-zero: 601 + 106)
• b = 1 → 727 (e.g. 611 + 116)
• b = 2 → 747 (e.g. 621 + 126)
• b = 3 → 767 (e.g. 631 + 136)
• b = 4 → 787 (e.g. 641 + 146) - Case B (With carry to hundreds): a + c = 6, plus 1 carried from the tens column.
The tens sum 2b must be ≥ 10, so b ranges from 5 to 9:
• b = 5 (2b = 10, carry 1, unit 6) → 716 (e.g. 551 + 155)
• b = 6 (2b = 12, carry 1, unit 6) → 736 (e.g. 561 + 165)
• b = 7 (2b = 14, carry 1, unit 6) → 756 (e.g. 571 + 175)
• b = 8 (2b = 16, carry 1, unit 6) → 776 (e.g. 581 + 185)
• b = 9 (2b = 18, carry 1, unit 6) → 796 (e.g. 591 + 195)
Complete list of 10 totals: 707, 716, 727, 736, 747, 756, 767, 776, 787, 796.
3. Totals Between 300–400 and 800–900
- Between 300 and 400: 10 totals (303, 312, 323, 332, 343, 352, 363, 372, 383, 392).
- Between 800 and 900: 10 totals (808, 817, 828, 837, 848, 857, 868, 877, 888, 897).
4. 4-Digit Totals (1200–1300 and 1251)
To get a 4-digit sum around 1200, a + c = 11 (since units digit must be 1 and sum exceeds 1000).
• Units column: 1 (with 1 carried).
• Tens column: 2b + 1.
• Hundreds/Thousands column: 11 + carry from tens.
For 1251: tens digit is 5, so 2b + 1 = 5 → 2b = 4 → b = 2 (no carry into hundreds).
With a + c = 11 and b = 2: e.g., 922 + 229 = 1251, 823 + 328 = 1251, 724 + 427 = 1251, 625 + 526 = 1251.
• All totals between 1200 and 1300: 10 totals (1211, 1231, 1251, 1271, 1291 without tens carry; and 1201, 1221, 1241, 1261, 1281 from a + c = 12 with carry).
• Between 1900 and 2000: 0 totals. The maximum possible sum of two 3-digit numbers is 999 + 999 = 1998, but for reversals the max is 998 + 899 = 1897. None fall in 1900–2000.
5. Subtraction Extension
Reversal subtraction for a 3-digit number is: (100a + 10b + c) − (100c + 10b + a) = 99(a − c).
Every 3-digit subtraction result must be a multiple of 99.
• Between 200 and 300: The only multiple of 99 in this range is 99 × 3 = 297 (e.g. 542 − 245 = 297). There is exactly 1 number.
• 4-digit subtraction between 1200 and 1300: For 4-digit numbers abcd − dcba = 999(a − d) + 90(b − c). The only values between 1200 and 1300 are 1269 and 1278 (fewer than 5, more than 1).