Puzzle for June 24, 2020 ( )
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 C with B + F (from eq.2): A – B = B + B + F – F which becomes A – B = 2×B Add B to both sides of the above equation: A – B + B = 2×B + B which makes A = 3×B
Hint #2
In eq.6, replace A with 3×B: E – F = 3×B ÷ B which becomes E – F = 3 Add F to both sides of the equation above: E – F + F = 3 + F which becomes eq.6a) E = 3 + F
Hint #3
In eq.5, substitute B + F for C (from eq.2), and 3×B for A: B + B + F + D – F = 3×B + F which becomes 2×B + D = 3×B + F Subtract 2×B from both sides of the equation above: 2×B + D – 2×B = 3×B + F – 2×B which becomes eq.5a) D = B + F
Hint #4
In eq.2, substitute D for B + F (from eq.5a): C = D
Hint #5
Substitute B + F for D (from eq.5a), 3 + F for E (from eq.6a), and 3×B for A in eq.3: B + F + 3 + F + F = 3×B + B which becomes B + 3×F + 3 = 4×B Subtract both B and 3 from each side of the equation above: B + 3×F + 3 – B – 3 = 4×B – B – 3 which becomes 3×F = 3×B – 3 Divide both sides by 3: 3×F ÷ 3 = (3×B – 3) ÷ 3 which makes eq.3a) F = B – 1
Hint #6
Substitute B – 1 for F (from eq.3a) in eq.5a: D = B + B – 1 which makes D = 2×B – 1 and also makes eq.5b) C = D = 2×B – 1
Hint #7
Substitute B – 1 for F (from eq.3a) in eq.6a: E = 3 + B – 1 which makes eq.6b) E = 2 + B
Solution
Substitute 3×B for A, 2×B – 1 for C and D (from eq.5b), 2 + B for E (from eq.6b), and B – 1 for F (from eq.3a) in eq.1: 3×B + B + 2×B – 1 + 2×B – 1 + 2 + B + B – 1 = 29 which simplifies to 10×B – 1 = 29 Add 1 to both sides of the equation above: 10×B – 1 + 1 = 29 + 1 which makes 10×B = 30 Divide each side by 10: 10×B ÷ 10 = 30 ÷ 10 which means B = 3 making A = 3×B = 3 × 3 = 9 C = D = 2×B – 1 = 2×3 – 1 = 6 – 1 = 5 (from eq.5b) E = 2 + B = 2 + 3 = 5 (from eq.6b) F = B – 1 = 3 – 1 = 2 (from eq.3a) and ABCDEF = 935552