Puzzle for January 31, 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
Add C to both sides of eq.2: C + C = B + D + F + C which is the same as eq.2a) 2×C = B + C + D + F Add D to both sides of eq.6: B + C + F + D = A + D + D which is the same as eq.6a) B + C + D + F = A + 2×D
Hint #2
In eq.6a, substitute 2×C for B + C + D + F (from eq.2a): 2×C = A + 2×D Subtract 2×D from each side of the above equation: 2×C – 2×D = A + 2×D – 2×D which becomes eq.6b) 2×C – 2×D = A
Hint #3
In eq.3, substitute 2×C – 2×D for A (from eq.6b): 2×C – 2×D = C + D In the above equation, add 2×D to each side, and subtract C from both sides: 2×C – 2×D + 2×D – C = C + D + 2×D – C which makes C = 3×D
Hint #4
Substitute 3×D for C in eq.3: A = 3×D + D which makes A = 4×D
Hint #5
Substitute 4×D for A in eq.4: E + F = 4×D – E Subtract E from both sides of the above equation: E + F – E = 4×D – E – E which becomes eq.4a) F = 4×D – 2×E
Hint #6
Substitute (4×D – 2×E) for F (from eq.4a), and 3×D for C in eq.5: D + E – (4×D – 2×E) = 3×D – E which is equivalent to D + E – 4×D + 2×E = 3×D – E which becomes –3×D + 3×E = 3×D – E Add E and 3×D to both sides of the equation above: –3×D + 3×E + E + 3×D = 3×D – E + E + 3×D which means 4×E = 6×D Divide both sides by 4: 4×E ÷ 4 = 6×D ÷ 4 which makes E = 1½×D
Hint #7
Substitute (1½×D) for E in eq.4a: F = 4×D – 2×(1½×D) which becomes F = 4×D – 3×D which makes F = D
Hint #8
Substitute 3×D for C, and D for F in eq.2: 3×D = B + D + D which becomes 3×D = B + 2×D Subtract 2×D from both sides of the above equation: 3×D – 2×D = B + 2×D – 2×D which makes D = B
Solution
Substitute 4×D for A, D for B and F, 3×D for C, and 1½×D for E in eq.1: 4×D + D + 3×D + D + 1½×D + D = 23 which simplifies to 11½×D = 23 Divide both sides of the equation above by 11½: 11½×D ÷ 11½ = 23 ÷ 11½ which means D = 2 making A = 4×D = 4 × 2 = 8 B = F = D = 2 C = 3×D = 3 × 2 = 6 E = 1½×D = 1½ × 2 = 3 and ABCDEF = 826232