Puzzle for November 11, 2019 ( )
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.2, replace E with B + C (from eq.5): B + B + C = A + C which becomes 2×B + C = A + C Subtract C from both sides of the equation above: 2×B + C – C = A + C – C which makes 2×B = A
Hint #2
In eq.4, replace F with A + E (from eq.6): D + E = A + A + E which becomes D + E = 2×A + E Subtract E from both sides of the equation above: D + E – E = 2×A + E – E which makes eq.4a) D = 2×A
Hint #3
In eq.4a, substitute (2×B) for A: D = 2×(2×B) which means D = 4×B
Hint #4
Substitute 2×B for A, and 4×B for D in eq.4a: C = 2×B + 4×B which makes C = 6×B
Hint #5
Substitute 2×B for A, and 6×B for C in eq.2: B + E = 2×B + 6×B which becomes B + E = 8×B Subtract B from both sides of the equation above: B + E – B = 8×B – B which makes E = 7×B
Hint #6
Substitute 2×B for A, and 7×B for E in eq.6: F = 2×B + 7×B which makes F = 9×B
Solution
Substitute 2×B for A, 6×B for C, 4×B for D, 7×B for E, and 9×B for F in eq.1: 2×B + B + 6×B + 4×B + 7×B + 9×B = 29 which simplifies to 29×B = 29 Divide both sides of the equation above by 29: 29×B ÷ 29 = 29 ÷ 29 which means B = 1 making A = 2×B = 2 × 1 = 2 C = 6×B = 6 × 1 = 6 D = 4×B = 4 × 1 = 4 E = 7×B = 7 × 1 = 7 F = 9×B = 9 × 1 = 9 and ABCDEF = 216479