Puzzle for August 5, 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.5, subtract D from both sides, and add A to both sides: A – D – D + A = B + D – A – D + A which becomes eq.5a) 2×A – 2×D = B
Hint #2
In eq.4, replace B with 2×A – 2×D (from eq.5a): 2×A – 2×D – A = A + E – F which becomes A – 2×D = A + E – F In the above equation, subtract A from both sides, and add F to both sides: A – 2×D – A + F = A + E – F – A + F which becomes eq.4a) –2×D + F = E
Hint #3
In eq.2, replace E with –2×D + F (from eq.4a): –2×D + F = D + F Subtract D and F from each side of the above equation: –2×D + F – F – D = D + F – F – D which makes –3×D = 0 which means D = 0
Hint #4
In eq.5a, substitute 0 for D: 2×A – 2×0 = B which becomes 2×A – 0 = B which makes eq.5b) 2×A = B
Hint #5
In eq.2, substitute 0 for D: E = 0 + F which makes E = F
Hint #6
In eq.3, substitute 0 for D: 0 + E = B + C which makes E = B + C and also makes eq.3a) F = E = B + C
Hint #7
eq.6 may be written as: B = (C + E + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × B = 3 × (C + E + F) ÷ 3 which becomes eq.6a) 3×B = C + E + F
Hint #8
Substitute B + C for E and F (from eq.3a) in eq.6a: 3×B = C + B + C + B + C which becomes 3×B = 3×C + 2×B Subtract 2×B from both sides of the above equation: 3×B – 2×B = 3×C + 2×B – 2×B which makes B = 3×C
Hint #9
Substitute 3×C for B in eq.3a: F = E = 3×C + C which makes F = E = 4×C
Hint #10
Substitute 3×C for B in eq.5b: 2×A = 3×C Divide both sides of the above equation by 2: 2×A ÷ 2 = 3×C ÷ 2 which makes A = 1½×C
Solution
Substitute 1½×C for A, 3×C for B, 0 for D, and 4×C for E and F in eq.1: 1½×C + 3×C + C + 0 + 4×C + 4×C = 27 which simplifies to 13½×C = 27 Divide both sides of the above equation by 13½: 13½×C ÷ 13½ = 27 ÷ 13½ which means C = 2 making A = 1½×C = 1½ × 2 = 3 B = 3×C = 3 × 2 = 6 E = F = 4×C = 4 × 2 = 8 and ABCDEF = 362088