Puzzle for December 14, 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.
* BC is a 2-digit number (not B×C).
Scratchpad
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Hint #1
In eq.6, add F to both sides, and subtract B from both sides: A + F + F – B = B + D – F + F – B which becomes A + 2×F – B = D In eq.2, replace D with A + 2×F – B: A + B = A + 2×F – B In the equation above, add B to both sides, and subtract A from each side: A + B + B – A = A + 2×F – B + B – A which simplifies to 2×B = 2×F Divide both sides by 2: 2×B = 2×F which means B = F
Hint #2
In eq.3, replace F with B: B + C = D – B Add B to each side of the above equation: B + C + B = D – B + B which becomes eq.3a) 2×B + C = D
Hint #3
In eq.3, replace D with A + B (from eq.2), and replace F with B: B + C = A + B – B which becomes eq.3b) B + C = A
Hint #4
Substitute (2×B + C) for D (from eq.3a), B + C for A (from eq.3b), and B for F in eq.4: E – (2×B + C) = B + C + B + C + (2×B + C) + B which becomes E – 2×B – C = 5×B + 3×C Add 2×B and C to both sides of the above equation: E – 2×B – C + 2×B + C = 5×B + 3×C + 2×B + C which simplifies to eq.4a) E = 7×B + 4×C
Hint #5
eq.5 may be written as: D + E + F = 10×B + C Substitute 2×B + C for D (from eq.3a), 7×B + 4×C for E (from eq.4a), and B for F in the above equation: 2×B + C + 7×B + 4×C + B = 10×B + C which becomes 10×B + 5×C = 10×B + C Subtract 10×B and C from each side of the equation above: 10×B + 5×C – 10×B – C = 10×B + C – 10×B – C which makes 4×C = 0 which means C = 0
Hint #6
Substitute 0 for C in eq.3b: B + 0 = A which means B = A
Hint #7
Substitute 0 for C in eq.3a: 2×B + 0 = D which makes 2×B = D
Hint #8
Substitute 0 for C in eq.4a: E = 7×B + 4×0 which means E = 7×B
Solution
Substitute B for A and F, 0 for C, 2×B for D, and 7×B for E in eq.1: B + B + 0 + 2×B + 7×B + B = 12 which simplifies to 12×B = 12 Divide both sides of the equation above by 12: 12×B ÷ 12 = 12 ÷ 12 which means B = 1 making A = F = B = 1 D = 2×B = 2 × 1 = 2 E = 7×B = 7 × 1 = 7 and ABCDEF = 110271