Puzzle for March 17, 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
eq.4 may be written as: A - C - D + F = B - A + C + E - F In the above equation, replace B - A with A + C - F (from eq.3): A - C - D + F = A + C - F + C + E - F which becomes A - C - D + F = A + 2×C - 2×F + E Subtract A from both sides, and add C and D and 2×F to both sides: A - C - D + F - A + C + D + 2×F = A + 2×C - 2×F + E - A + C + D + 2×F which becomes eq.4a) 3×F = 3×C + E + D
Hint #2
eq.6 may be written as: F = (C + D + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × F = 3 × (C + D + E) ÷ 3 which becomes eq.6a) 3×F = C + D + E
Hint #3
In eq.6a, replace 3×F with 3×C + E + D (from eq.4a): 3×C + E + D = C + D + E Subtract C, E, and D from each side of the above equation: 3×C + E + D - C - E - D = C + D + E - C - E - D which simplifies to 2×C = 0 which means C = 0
Hint #4
eq.5 may be written as: B = (D + E + F) ÷ 3 Multiply both sides of the equation above by 3: 3 × B = 3 × (D + E + F) ÷ 3 which becomes eq.5a) 3×B = D + E + F
Hint #5
Add F to both sides of eq.6a: 3×F + F = C + D + E + F which becomes eq.6b) 4×F = C + D + E + F
Hint #6
In eq.6b, substitute 0 for C, and 3×B for D + E + F (from eq.5a): 4×F = 0 + 3×B which makes 4×F = 3×B Divide both sides of the above equation by 4: 4×F ÷ 4 = 3×B ÷ 4 which makes F = ¾×B
Hint #7
Substitute 0 for C, and ¾×B for F in eq.3: B - A = A + 0 - ¾×B which becomes B - A = A - ¾×B Add A and ¾×B to both sides of the above equation: B - A + A + ¾×B = A - ¾×B + A + ¾×B which makes 1¾×B = 2×A Divide both sides by 2: 1¾×B ÷ 2 = 2×A ÷ 2 which makes ⅞×B = A
Hint #8
Substitute ⅞×B for A in eq.2: ⅞×B - B = B - D which becomes -⅛×B = B - D Add ⅛×B and D to both sides of the equation above: -⅛×B + ⅛×B + D = B - D + ⅛×B + D which makes D = 1⅛×B
Hint #9
Substitute (¾×B) for F, 0 for C, and 1⅛×B for D in eq.6a: 3×(¾×B) = 0 + 1⅛×B + E which becomes 2¼×B = 1⅛×B + E Subtract 1⅛×B from each side of the equation above: 2¼×B - 1⅛×B = 1⅛×B + E - 1⅛×B which makes 1⅛×B = E
Solution
Substitute ⅞×B for A, 0 for C, and 1⅛×B for D and E, and ¾×B for F in eq.1: ⅞×B + B + 0 + 1⅛×B + 1⅛×B + ¾×B = 39 which simplifies to 4⅞×B = 39 Divide both sides of the above equation by 4⅞: 4⅞×B ÷ 4⅞ = 39 ÷ 4⅞ which means B = 8 making A = ⅞×B = ⅞ × 8 = 7 D = E = 1⅛×B = 1⅛ × 8 = 9 F = ¾×B = ¾ × 8 = 6 and ABCDEF = 780996