Puzzle for February 27, 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.6, replace C with B + D (from eq.4): B + D – E = D + E In the above equation, add E to each side, and subtract D from both sides: B + D – E + E – D = D + E + E – D which simplifies to eq.6a) B = 2×E
Hint #2
Add C and F to both sides of eq.3: D – F + C + F = E – C + C + F which becomes eq.3a) D + C = E + F In eq.4, replace B with 2×E: eq.4a) 2×E + D = C
Hint #3
In eq.3a, substitute 2×E + D for C (from eq.4a): D + 2×E + D = E + F Subtract E from both sides of the above equation: D + 2×E + D – E = E + F – E which becomes eq.3b) 2×D + E = F
Hint #4
Substitute 2×E for B, and 2×D + E for F (from eq.3b) in eq.5: A – D = 2×E + 2×D + E which becomes A – D = 2×D + 3×E Add D to both sides of the equation above: A – D + D = 2×D + 3×E + D which becomes eq.5a) A = 3×D + 3×E
Hint #5
Substitute 3×D + 3×E for A (from eq.5a), 2×E for B, and 2×E + D for C (from eq.4a) in eq.2: 3×D + 3×E = 2×E + 2×E + D which becomes 3×D + 3×E = D + 4×E Subtract D and 3×E from both sides of the above equation: 3×D + 3×E – D + 4×E = D + 4×E – D – 4×E which makes 2×D = E
Hint #6
Substitute (2×D) for E in eq.5a: A = 3×D + 3×(2×D) which becomes A = 3×D + 6×D which makes A = 9×D
Hint #7
Substitute (2×D) for E in eq.6a: B = 2×(2×D) which makes B = 4×D
Hint #8
Substitute (2×D) for E in eq.4a: 2×(2×D) + D = C which becomes 4×D + D = C which makes 5×D = C
Hint #9
Substitute 2×D for E in eq.3b: 2×D + 2×D = F which makes 4×D = F
Solution
Substitute 9×D for A, 4×D for B and F, 5×D for C, and 2×D for E in eq.1: 9×D + 4×D + 5×D + D + 2×D + 4×D = 25 which simplifies to 25×D = 25 Divide both sides of the equation above by 25: 25×D ÷ 25 = 25 ÷ 25 which means D = 1 making A = 9×D = 9 × 1 = 9 B = F = 4×D = 4 × 1 = 4 C = 5×D = 5 × 1 = 5 E = 2×D = 2 × 1 = 2 and ABCDEF = 945124