Puzzle for January 6, 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 E to both sides of eq.3: F - E + E = E - A + E which becomes eq.3a) F = 2×E - A
Hint #2
In eq.2, substitute 2×E - A for F (from eq.3a): C + 2×E - A = A + E In the above equation, subtract 2×E from both sides, and add A to both sides: C + 2×E - A - 2×E + A = A + E - 2×E + A which becomes eq.2a) C = 2×A - E
Hint #3
Add A to both sides of eq.4: D - A + A = A - E + A which becomes D = 2×A - E Combining eq.2a with the above equation yields: eq.4a) C = D = 2×A - E
Hint #4
In eq.5, substitute 2×A - E for C and D (from eq.4a), and 2×E - A for F (from eq.3a): B + 2×A - E + 2×A - E = A - B + E + 2×E - A which becomes B + 4×A - 2×E = -B + 3×E In the above equation, add 2×E and B to both sides, and subtract 4×A from both sides: B + 4×A - 2×E + 2×E + B - 4×A = -B + 3×E + 2×E + B - 4×A which simplifies to 2×B = 5×E - 4×A Divide both sides by 2: 2×B ÷ 2 = (5×E - 4×A) ÷ 2 which makes eq.5a) B = 2½×E - 2×A
Hint #5
eq.6 may be written as: A = (B + C + D + E + F) ÷ 5 Multiply both sides of the above equation by 5: 5 × A = 5 × (B + C + D + E + F) ÷ 5 which becomes eq.6a) 5×A = B + C + D + E + F
Hint #6
In eq.6a, substitute 2½×E - 2×A for B (from eq.5a), 2×A - E for C and D (from eq.4a), and 2×E - A for F (from eq.3a): 5×A = 2½×E - 2×A + 2×A - E + 2×A - E + E + 2×E - A which becomes 5×A = 3½×E + A Subtract A from each side of the equation above: 5×A - A = 3½×E + A - A which makes 4×A = 3½×E Divide both sides by 4: 4×A ÷ 4 = 3½×E ÷ 4 which makes A = ⅞×E
Hint #7
Substitute (⅞×E) for A in eq.5a: B = 2½×E - 2×(⅞×E) which becomes B = 2½×E - 1¾×E which makes B = ¾×E
Hint #8
Substitute (⅞×E) for A in eq.4a: C = D = 2×(⅞×E) - E which becomes C = D = 1¾×E - E which makes C = D = ¾×E
Hint #9
Substitute ⅞×E for A in eq.3a: F = 2×E - ⅞×E which makes F = 1⅛×E
Solution
Substitute ⅞×E for A, ¾×E for B and C and D, and 1⅛×E for F in eq.1: ⅞×E + ¾×E + ¾×E + ¾×E + E + 1⅛×E = 42 which simplifies to 5¼×E = 42 Divide both sides of the above equation by 5¼: 5¼×E ÷ 5¼ = 42 ÷ 5¼ which means E = 8 making A = ⅞×E = ⅞ × 8 = 7 B = C = D = ¾×E = ¾ × 8 = 6 F = 1⅛×E = 1⅛ × 8 = 9 and ABCDEF = 766689