Puzzle for January 21, 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.3, substitute (C + F) for B (from eq.2): E – F = A – (C + F) + C which is equivalent to E – F = A – C – F + C which becomes E – F = A – F Add F to both sides of the equation above: E – F + F = A – F + F which makes E = A
Hint #2
In eq.4, replace E with A: C – D = A – A which becomes C – D = 0 Add D to each side of the above equation: C – D + D = 0 + D which makes D = C
Hint #3
In eq.5, replace D with C: F – C = C + C which becomes F – C = 2×C Add C to both sides of the equation above: F – C + C = 2×C + C which makes F = 3×C
Hint #4
In eq.2, substitute 3×C for F: B = C + 3×C which makes B = 4×C
Hint #5
Substitute A for E, 4×C for B, C for D, and 3×C for F in eq.6: A + A = 4×C + C + C + 3×C which makes 2×A = 9×C Divide both sides by 2: 2×A ÷ 2 = 9×C ÷ 2 which means A = 4½×C and also means E = A = 4½×C
Solution
Substitute 4½×C for A and E, 4×C for B, C for D, and 3×C for F in eq.1: 4½×C + 4×C + C + C + 4½×C + 3×C = 36 which simplifies to 18×C = 36 Divide both sides of the above equation by 18: 18×C ÷ 18 = 36 ÷ 18 which means C = 2 making A = E = 4½×C = 4½ × 2 = 9 B = 4×C = 4 × 2 = 8 D = C = 2 F = 3×C = 3 × 2 = 6 and ABCDEF = 982296