Puzzle for December 6, 2021 ( )
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 E with A + C (from eq.3), and F with B + C (from eq.2): A + C + B + C = A + B + D which becomes A + 2×C + B = A + B + D Subtract A and B from both sides of the above equation: A + 2×C + B – A – B = A + B + D – A – B which makes 2×C = D
Hint #2
In eq.4, replace D with 2×C: B = C + 2×C which makes B = 3×C
Hint #3
In eq.2, substitute 3×C for B: F = 3×C + C which makes F = 4×C
Hint #4
Substitute 4×C for F, and 3×C for B in eq.5: C + 4×C = A – 3×C which becomes 5×C = A – 3×C Add 3×C to both sides of the above equation: 5×C + 3×C = A – 3×C + 3×C which makes 8×C = A
Hint #5
Substitute 8×C for A in eq.3: E = 8×C + C which makes E = 9×C
Solution
Substitute 8×C for A, 3×C for B, 2×C for D, 9×C for E, and 4×C for F in eq.1: 8×C + 3×C + C + 2×C + 9×C + 4×C = 27 which simplifies to 27×C = 27 Divide both sides of the above equation by 27: 27×C ÷ 27 = 27 ÷ 27 which means C = 1 making A = 8×C = 8 × 1 = 8 B = 3×C = 3 × 1 = 3 D = 2×C = 2 × 1 = 2 E = 9×C = 9 × 1 = 9 F = 4×C = 4 × 1 = 4 and ABCDEF = 831294