Puzzle for March 4, 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
Add F to both sides of eq.5: B + E + F = A + C - F + F which becomes eq.5a) B + E + F = A + C Add E, A, and C to both sides of eq.6: A - D - E + E + A + C = B + F - A - C + E + A + C which becomes 2×A - D + C = B + F + E which may be written as eq.6a) 2×A + C - D = B + E + F
Hint #2
In eq.5a, replace B + E + F with 2×A + C - D (from eq.6a): 2×A + C - D = A + C In the above equation, add D to both sides, and subtract A and C from both sides: 2×A + C - D + D - A - C = A + C + D - A - C which simplifies to A = D
Hint #3
In eq.2, replace A with D: B + D = D + F Subtract D from each side of the equation above: B + D - D = D + F - D which makes B = F
Hint #4
In eq.3, replace A with D: C + E = D + D which becomes C + E = 2×D Subtract E from each side of the above equation: C + E - E = 2×D - E which becomes eq.3a) C = 2×D - E
Hint #5
In eq.4, substitute B for F: D - B = B - E + B which becomes D - B = 2×B - E Add B and E to both sides of the equation above: D - B + B + E = 2×B - E + B + E which becomes eq.4a) D + E = 3×B
Hint #6
Substitute D for A, and B for F in eq.6a: 2×D + C - D = B + E + B which becomes D + C = 2×B + E Multiply both sides of the above equation by 1½: 1½×(D + C) = 1½×(2×B + E) which becomes eq.6b) 1½×D + 1½×C = 3×B + 1½×E
Hint #7
Substitute 2×D - E for C (from eq.3a), and D + E for 3×B (from eq.4a) in eq.6b: 1½×D + 1½×(2×D - E) = D + E + 1½×E which becomes 1½×D + 3×D - 1½×E = D + 2½×E which becomes 4½×D - 1½×E = D + 2½×E In the above equation, add 1½×E to both sides, and subtract D from both sides: 4½×D - 1½×E + 1½×E - D = D + 2½×E + 1½×E - D which simplifies to 3½×D = 4×E Divide both sides by 4: 3½×D ÷ 4 = 4×E ÷ 4 which makes ⅞×D = E
Hint #8
Substitute ⅞×D for E in eq.4a: D + ⅞×D = 3×B which makes 1⅞×D = 3×B Divide both sides of the above equation by 3: 1⅞×D ÷ 3 = 3×B ÷ 3 which makes ⅝×D = B and also makes F = B = ⅝×D
Hint #9
Substitute ⅞×D for E in eq.3a: C = 2×D - ⅞×D which makes C = 1⅛×D
Solution
Substitute D for A, ⅝×D for B and F, 1⅛×D for C, and ⅞×D for E in eq.1: D + ⅝×D + 1⅛×D + D + ⅞×D + ⅝×D = 42 which simplifies to 5¼×D = 42 Divide both sides of the above equation by 5¼: 5¼×D ÷ 5¼ = 42 ÷ 5¼ which means D = 8 making A = D = 8 B = F = ⅝×D = ⅝ × 8 = 5 C = 1⅛×D = 1⅛ × 8 = 9 E = ⅞×D = ⅞ × 8 = 7 and ABCDEF = 859875