Puzzle for July 22, 2022 ( )
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 E + F with A + B + C (from eq.6): C + D = A + B + C – D In the above equation, subtract C from both sides, and add D to both sides: C + D – C + D = A + B + C – D – C + D which becomes eq.4a) 2×D = A + B
Hint #2
In eq.6, replace E with A + C + D (from eq.2): A + C + D + F = A + B + C Subtract A and C from each side of the equation above: A + C + D + F – A – C = A + B + C – A – C which simplifies to eq.6a) D + F = B
Hint #3
In eq.4a, substitute D + F for B (from eq.6a): 2×D = A + D + F Subtract D from each side of the above equation: 2×D – D = A + D + F – D which becomes eq.4b) D = A + F
Hint #4
Substitute A + F for D (from eq.4b) in eq.3: C + F = A – C + A + F which becomes C + F = 2×A – C + F In the above equation, subtract F from both sides, and add C to both sides: C + F – F + C = 2×A – C + F – F + C which makes 2×C = 2×A Divide both sides by 2: 2×C ÷ 2 = 2×A ÷ 2 which makes C = A
Hint #5
Substitute A for C in eq.2: E = A + A + D which becomes eq.2a) E = 2×A + D
Hint #6
Substitute A for C, and 2×A + D for E (from eq.2a) into eq.5: B + D = A + A + 2×A + D which becomes B + D = 4×A + D Subtract D from both sides of the above equation: B + D – D = 4×A + D – D which makes B = 4×A
Hint #7
Substitute 4×A for B in eq.4a: 2×D = A + 4×A which makes 2×D = 5×A Divide both sides of the above equation by 2: 2×D ÷ 2 = 5×A ÷ 2 which makes D = 2½×A
Hint #8
Substitute 2½×A for D in eq.4b: 2½×A = A + F Subtract A from both sides of the equation above: 2½×A – A = A + F – A which makes 1½×A = F
Hint #9
Substitute 2½×A for D in eq.2a: E = 2×A + 2½×A which makes E = 4½×A
Solution
Substitute 4×A for B, A for C, 2½×A for D, 4½×A for E, and 1½×A for F in eq.1: A + 4×A + A + 2½×A + 4½×A + 1½×A = 29 which simplifies to 14½×A = 29 Divide both sides of the above equation by 14½: 14½×A ÷ 14½ = 29 ÷ 14½ which means A = 2 making B = 4×A = 4 × 2 = 8 C = A = 2 D = 2½×A = 2½ × 2 = 5 E = 4½×A = 4½ × 2 = 9 F = 1½×A = 1½ × 2 = 3 and ABCDEF = 282593