Puzzle for August 17, 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.4, replace D + F with A + C (from eq.2): C + E = A + B + A + C Subtract C from both sides of the equation above: C + E – C = A + B + A + C – C which becomes eq.4a) E = 2×A + B
Hint #2
In eq.4a, replace E with A + D (from eq.3): A + D = 2×A + B Subtract A from both sides of the above equation: A + D – A = 2×A + B – A which becomes eq.4b) D = A + B
Hint #3
In eq.5, replace E with 2×A + B (from eq.4a): A – B + C = B + 2×A + B which becomes A – B + C = 2×A + 2×B In the above equation, add B to both sides, and subtract A from each side: A – B + C + B – A = 2×A + 2×B + B – A which becomes eq.5a) C = A + 3×B
Hint #4
In eq.2, substitute A + B for D (from eq.4b), and A + 3×B for C (from eq.5a): A + B + F = A + A + 3×B which becomes A + B + F = 2×A + 3×B Subtract A and B from both sides of the equation above: A + B + F – A – B = 2×A + 3×B – A – B which simplifies to eq.2a) F = A + 2×B
Hint #5
Substitute A + 3×B for C (from eq.5a), A + 2×B for F (from eq.2a), A + B for D (from eq.4b), and 2×A + B for E (from eq.4a) in eq.6: B + A + 3×B + A + 2×B = A + B + 2×A + B which becomes 2×A + 6×B = 3×A + 2×B Subtract 2×A and 2×B from both sides: 2×A + 6×B – 2×A – 2×B = 3×A + 2×B – 2×A – 2×B which simplifies to 4×B = A
Hint #6
Substitute 4×B for A in eq.5a: C = 4×B + 3×B which makes C = 7×B
Hint #7
Substitute 4×B for A in eq.4b: D = 4×B + B which makes D = 5×B
Hint #8
Substitute (4×B) for A in eq.4a: E = 2×(4×B) + B which is equivalent to E = 8×B + B which means E = 9×B
Hint #9
Substitute 4×B for A in eq.2a: F = 4×B + 2×B which makes F = 6×B
Solution
Substitute 4×B for A, 7×B for C, 5×B for D, 9×B for E, 6×B for F in eq.1: 4×B + B + 7×B + 5×B + 9×B + 6×B = 32 which simplifies to 32×B = 32 Divide both sides of the equation above by 32: 32×B ÷ 32 = 32 ÷ 32 which means B = 1 making A = 4×B = 4 × 1 = 4 C = 7×B = 7 × 1 = 7 D = 5×B = 5 × 1 = 5 E = 9×B = 9 × 1 = 9 F = 6×B = 6 × 1 = 6 and ABCDEF = 417596