Puzzle for July 7, 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
Subtract B from each side of eq.3: A – B = B + D + E – B which becomes eq.3a) A – B = D + E In eq.6, replace D + E with A – B (from eq.3a): C + A – B = A – C – E + F In the above equation, subtract A from both sides, and add B and C and E to both sides: C + A – B – A + B + C + E = A – C – E + F – A + B + C + E which simplifies to eq.6a) 2×C + E = B + F
Hint #2
In eq.6a, replace E with B + C (from eq.2): 2×C + B + C = B + F which becomes 3×C + B = B + F Subtract B from each side of the equation above: 3×C + B – B = B + F – B which makes eq.6b) 3×C = F
Hint #3
In eq.4, substitute 3×C for F (from eq.6b): 3×C – C = C + D – B which becomes 2×C = C + D – B In the equation above, subtract C from both sides, and add B to both sides: 2×C – C + B = C + D – B – C + B which becomes C + B = D which may be written as eq.4a) B + C = D
Hint #4
Substitute E for B + C (from eq.2) into eq.4a: E = D
Hint #5
Substitute (B + C) for D (from eq.4a) and for E (from eq.2), and 3×C for F (from eq.6b) in eq.5: (B + C) + (B + C) = C – (B + C) + 3×C which becomes B + C + B + C = C – B – C + 3×C which becomes 2×B + 2×C = 3×C – B In the equation above, subtract 2×C from both sides, and add B to both sides: 2×B + 2×C – 2×C + B = 3×C – B – 2×C + B which makes 3×B = C
Hint #6
Substitute 3×B for C in eq.4a: B + 3×B = D which makes 4×B = D and also makes 4×B = D = E
Hint #7
Substitute 3×B for C in eq.6b: 3×(3×B) = F which makes 9×B = F
Hint #8
Substitute 4×B for both D and E in eq.3: A = B + 4×B + 4×B which makes A = 9×B
Solution
Substitute 9×B for A and F, 3×B for C, and 4×B for D and E in eq.1: 9×B + B + 3×B + 4×B + 4×B + 9×B = 30 which simplifies to 30×B = 30 Divide both sides of the above equation by 30: 30×B ÷ 30 = 30 ÷ 30 which means B = 1 making A = F = 9×B = 9 × 1 = 9 C = 3×B = 3 × 1 = 3 D = E = 4×B = 4 × 1 = 4 and ABCDEF = 913449