Puzzle for March 16, 2023 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
Help Area
Hint #1
Add B to both sides of eq.5: B + C + D + B = A - B + F + B which becomes eq.5a) 2×B + C + D = A + F Add B and D to both sides of eq.4: A - B + B + D = B + C - D + B + D which becomes eq.4a) A + D = 2×B + C
Hint #2
In eq.5a, replace 2×B + C with A + D (from eq.4a): A + D + D = A + F which becomes A + 2×D = A + F Subtract A from each side of the equation above: A + 2×D - A = A + F - A which makes 2×D = F
Hint #3
In eq.2, replace F with 2×D: 2×D = D + E Subtract D from each side of the above equation: 2×D - D = D + E - D which makes D = E
Hint #4
In eq.6, substitute D for E, and 2×D for F: D + D + 2×D = A + B + C which becomes eq.6a) 4×D = A + B + C
Hint #5
Substitute 4×D for A + B + C (from eq.6a), D for E, and 2×D for F in eq.1: 4×D + D + D + 2×D = 32 which becomes 8×D = 32 Divide both sides of the above equation by 8: 8×D ÷ 8 = 32 ÷ 8 which makes D = 4 and makes E = D = 4 F = 2×D = 2 × 4 = 8
Hint #6
Substitute 8 for F, and 4 for E in eq.3: 8 - B = B - 4 Add B and 4 to both sides of the above equation: 8 - B + B + 4 = B - 4 + B + 4 which becomes 12 = 2×B Divide both sides by 2: 12 ÷ 2 = 2×B ÷ 2 which makes 6 = B
Hint #7
Substitute 6 for B, 4 for D, and 8 for F in eq.5a: 2×6 + C + 4 = A + 8 which becomes 12 + C + 4 = A + 8 which becomes 16 + C = A + 8 Subtract 8 from both sides of the above equation: 16 + C - 8 = A + 8 - 8 which makes eq.5b) 8 + C = A
Hint #8
Substitute 4 for D, 8 + C for A (from eq.5b), and 6 for B in eq.6a: 4×4 = 8 + C + 6 + C which becomes 16 = 14 + 2×C Subtract 14 from both sides of the above equation: 16 - 14 = 14 + 2×C - 14 which makes 2 = 2×C Divide both sides by 2: 2 ÷ 2 = 2×C ÷ 2 which makes 1 = C
Solution
Since C = 1, then: A = 8 + C = 8 + 1 = 9 (from eq.5b) and ABCDEF = 961448