Puzzle for October 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 to both sides of eq.3: F – D + D = A + D which becomes F = A + D In eq.5, replace F with A + D: A + C + E = B + A + D Subtract A from each side of the above equation: A + C + E – A = B + A + D – A which becomes eq.5a) C + E = B + D
Hint #2
Subtract D and E from both sides of eq.2: C + D – D – E = B + E – D – E which becomes C – E = B – D Add the left and right sides of the equation above to the left and right sides of eq.5a, respectively: C + E + C – E = B + D + B – D which becomes 2×C = 2×B Divide both sides of the equation above by 2: 2×C ÷ 2 = 2×B ÷ 2 which makes C = B
Hint #3
In eq.2, replace C with B: B + D = B + E Subtract B from both sides of the equation above: B + D – B = B + E – B which makes D = E
Hint #4
In eq.6, substitute B for C, D for E, and F – D for A (from eq.3): B + B + D + D = F – D – D + F which becomes 2×B + 2×D = 2×F – 2×D Add 2×D to both sides of the equation above: 2×B + 2×D + 2×D = 2×F – 2×D + 2×D which becomes 2×B + 4×D = 2×F Divide both sides by 2: (2×B + 4×D) ÷ 2 = 2×F ÷ 2 which becomes eq.6a) B + 2×D = F
Hint #5
Substitute D for E, F – D for A (from eq.3), and B for C in eq.4: D + D + F = F – D + B + B – F which becomes 2×D + F = –D + 2×B Subtract 2×D from both sides of the equation above: 2×D + F – 2×D = –D + 2×B – 2×D which becomes eq.4a) F = 2×B – 3×D
Hint #6
Substitute B + 2×D for F (from eq.6a) in eq.4a: B + 2×D = 2×B – 3×D In the equation above, subtract B from both sides, and add 3×D to both sides: B + 2×D – B + 3×D = 2×B – 3×D – B + 3×D which makes 5×D = B and also makes C = B = 5×D
Hint #7
Substitute 5×D for B in eq.6a: 5×D + 2×D = F which makes 7×D = F
Hint #8
Substitute 7×D for F in eq.3: 7×D – D = A which makes 6×D = A
Solution
Substitute 6×D for A, 5×D for B and C, D for E, and 7×D for F in eq.1: 6×D + 5×D + 5×D + D + D + 7×D = 25 which simplifies to 25×D = 25 Divide both sides of the above equation by 25: 25×D ÷ 25 = 25 ÷ 25 which means D = 1 making A = 6×D = 6 × 1 = 6 B = C = 5×D = 5 × 1 = 5 E = D = 1 F = 7×D = 7 × 1 = 7 and ABCDEF = 655117