Puzzle for June 17, 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.3: F + D + E = A + B – D – E + D + E which becomes F + D + E = A + B which is the same as D + E + F = A + B In eq.4, replace D + E + F with A + B: A + B = A + C Subtract A from both sides of the above equation: A + B – A = A + C – A which makes B = C
Hint #2
In eq.2, replace C with B: B + E = B + D Subtract B from each side of the above equation: B + E – B = B + D – B which becomes E = D
Hint #3
In eq.5, substitute D for E, and B for C: F – D – D = B + B – F which becomes F – 2×D = 2×B – F Add 2×D and F to both sides of the above equation: F – 2×D + 2×D + F = 2×C – F + 2×D + F which becomes 2×F = 2×C + 2×D Divide both sides by 2: 2×F ÷ 2 = (2×C + 2×D) ÷ 2 which becomes eq.5a) F = C + D
Hint #4
Substitute D for E, and C + D for F (from eq.5a) in eq.4: D + D + C + D = A + C which becomes 3×D + C = A + C Subtract C from each side of the above equation: 3×D + C – C = A + C – C which makes 3×D = A
Hint #5
Substitute 3×D for A, and C + D for F (from eq.5a) in eq.6: B + C – D = 3×D + D + C + D which becomes B + C – D = 5×D + C In the equation above, subtract C from each side, and add D to each side: B + C – D – C + D = 5×D + C – C + D which makes B = 6×D and also makes C = B = 6×D
Hint #6
Substitute 6×D for C in eq.5a: F = 6×D + D which makes F = 7×D
Solution
Substitute 3×D for A, 6×D for B and C, D for E, and 7×D for F in eq.1: 3×D + 6×D + 6×D + D + D + 7×D = 24 which simplifies to 24×D = 24 Divide both sides of the equation above by 24: 24×D ÷ 24 = 24 ÷ 24 which means D = 1 making A = 3×D = 3 × 1 = 3 B = C = 6×D = 6 × 1 = 6 E = D = 1 F = 7×D = 7 × 1 = 7 and ABCDEF = 366117