Puzzle for May 1, 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 D and E to both sides of eq.6: F – D + D + E = A – E + D + E which becomes F + E = A + D which is the same as eq.6a) E + F = A + D In eq.2, replace E + F with A + D (from eq.6a): D + A + D = A + B which becomes A + 2×D = A + B Subtract A from each side of the above equation: A + 2×D – A = A + B – A which makes eq.2a) 2×D = B
Hint #2
In eq.4, replace B with 2×D: 2×D – A = C + D Subtract D from both sides of the equation above: 2×D – A – D = C + D – D which becomes eq.4a) D – A = C
Hint #3
Add F to both sides of eq.5: C + F + F = A – F + F which becomes eq.5a) C + 2×F = A In eq.4a, substitute (C + 2×F) for A (from eq.5a): D – (C + 2×F) = C which becomes D – C – 2×F = C Add C and 2×F to each side of the equation above: D – C – 2×F + C + 2×F = C + C + 2×F which becomes eq.4b) D = 2×C + 2×F
Hint #4
Substitute C + 2×F for A (from eq.5a), and 2×C + 2×F for D (from eq.4b) in eq.6a: E + F = C + 2×F + 2×C + 2×F which becomes E + F = 3×C + 4×F Subtract F from both sides of the above equation: E + F – F = 3×C + 4×F – F which becomes eq.6b) E = 3×C + 3×F
Hint #5
Substitute 3×C + 3×F for E (from eq.6b), and 2×C + 2×F for D (from eq.4b) in eq.3: 3×C + 3×F – F = 2×C + 2×F – C which becomes 3×C + 2×F = C + 2×F Subtract C and 2×F from both sides of the equation above: 3×C + 2×F – C – 2×F = C + 2×F – C – 2×F which makes 2×C = 0 which means C = 0
Hint #6
Substitute 0 for C in eq.5a: 0 + 2×F = A which makes 2×F = A
Hint #7
Substitute 0 for C in eq.6b: E = 3×0 + 3×F which makes E = 3×F
Hint #8
Substitute 0 for C in eq.4b: D = 2×0 + 2×F which makes D = 2×F
Hint #9
Substitute (2×F) for D in eq.2a: 2×(2×F) = B which makes 4×F = B
Solution
Substitute 2×F for A and D, 4×F for B, 0 for C, and 3×F for E in eq.1: 2×F + 4×F + 0 + 2×F + 3×F + F = 24 which simplifies to 12×F = 24 Divide both sides of the equation above by 12: 12×F ÷ 12 = 24 ÷ 12 which means F = 2 making A = D = 2×F = 2 × 2 = 4 B = 4×F = 4 × 2 = 8 E = 3×F = 3 × 2 = 6 and ABCDEF = 480462