Puzzle for March 21, 2023 ( )
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.6, replace B + C - F with A - C + D (from eq.5): A - C + D = D + E + F - A - C In the above equation, add A and C to both sides, and subtract D from both sides: A - C + D + A + C - D = D + E + F - A - C + A + C - D which becomes eq.6a) 2×A = E + F
Hint #2
Add F and A to both sides of eq.4: A - F + F + A = B + E - A + F + A which becomes eq.4a) 2×A = B + E + F
Hint #3
In eq.4a, replace E + F with 2×A (from eq.6a): 2×A = B + 2×A Subtract 2×A from both sides of the equation above: 2×A - 2×A = B + 2×A - 2×A which makes B = 0
Hint #4
In eq.3, substitute 0 for B: E - D = 0 + D which becomes E - D = D Add D to both sides of the above equation: E - D + D = D + D which makes E = 2×D
Hint #5
Substitute 0 for B, and 2×D for E in eq.2: D + F = 0 + 2×D which becomes D + F = 2×D Subtract D from each side of the equation above: D + F - D = 2×D - D which makes F = D
Hint #6
Substitute 2×D for E, and D for F in eq.6a: 2×A = 2×D + D which becomes 2×A = 3×D Divide both sides of the above equation by 2: 2×A ÷ 2 = 3×D ÷ 2 which makes A = 1½×D
Hint #7
Substitute 0 for B, D for F, and 1½×D for A in eq.5: 0 + C - D = 1½×D - C + D which becomes C - D = 2½×D - C Add D and C to both sides of the equation above: C - D + D + C = 2½×D - C + D + C which makes 2×C = 3½×D Divide both sides by 2: 2×C ÷ 2 = 3½×D ÷ 2 which makes C = 1¾×D
Solution
Substitute 1½×D for A, 0 for B, 1¾×D for C, 2×D for E, and D for F in eq.1: 1½×D + 0 + 1¾×D + D + 2×D + D = 29 which simplifies to 7¼×D = 29 Divide both sides of the above equation by 7¼: 7¼×D ÷ 7¼ = 29 ÷ 7¼ which means D = 4 making A = 1½×D = 1½ × 4 = 6 C = 1¾×D = 1¾ × 4 = 7 E = 2×D = 2 × 4 = 8 F = D = 4 and ABCDEF = 607484