Puzzle for March 2, 2022 ( )
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
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Hint #1
In eq.6, replace F with B + C (from eq.2): C + B + C = E – C which becomes 2×C + B = E – C Add C to both sides of the above equation: 2×C + B + C = E – C + C which becomes eq.6a) 3×C + B = E
Hint #2
In eq.3, replace E with 3×C + B (from eq.6a): A = C + 3×C + B which becomes eq.3a) A = 4×C + B
Hint #3
In eq.4, substitute 4×C + B for A (from eq.3a): 4×C + B = B + C + D Subtract B and C from both sides of the equation above: 4×C + B – B – C = B + C + D – B – C which makes 3×C = D
Hint #4
Substitute 3×C for D, 3×C + B for E (from eq.6a), and B + C for F (from eq.2) in eq.5: 3×C + 3×C + B = B + B + C which becomes 6×C + B = 2×B + C Subtract B and C from each side of the equation above: 6×C + B – B – C = 2×B + C – B – C which makes 5×C = B
Hint #5
Substitute 5×C for B in eq.3a: A = 4×C + 5×C which makes A = 9×C
Hint #6
Substitute 5×C for B in eq.6a: 3×C + 5×C = E which makes 8×C = E
Hint #7
Substitute 5×C for B in eq.2: F = 5×C + C which makes F = 6×C
Solution
Substitute 9×C for A, 5×C for B, 3×C for D, 8×C for E, and 6×C for F in eq.1: 9×C + 5×C + C + 3×C + 8×C + 6×C = 32 which simplifies to 32×C = 32 Divide both sides of the above equation by 32: 32×C ÷ 32 = 32 ÷ 32 which means C = 1 making A = 9×C = 9 × 1 = 9 B = 5×C = 5 × 1 = 5 D = 3×C = 3 × 1 = 3 E = 8×C = 8 × 1 = 8 F = 6×C = 6 × 1 = 6 and ABCDEF = 951386