Puzzle for September 20, 2022 ( )
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.3, replace D with C + F (from eq.2): C + E = C + F + F which becomes C + E = C + 2×F Subtract C from each side of the equation above: C + E - C = C + 2×F - C which makes E = 2×F
Hint #2
eq.6 may be written as: E = (C + D + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × E = 3 × (C + D + F) ÷ 3 which becomes 3×E = C + D + F which may be written as eq.6a) 3×E = D + C + F
Hint #3
In eq.6a, replace E with (2×F), and C + F with D (from eq.2): 3×(2×F) = D + D which becomes 6×F = 2×D Divide both sides of the above equation by 2: 6×F ÷ 2 = 2×D ÷ 2 which makes 3×F = D
Hint #4
In eq.3, substitute 2×F for E, and 3×F for D: C + 2×F = 3×F + F which becomes C + 2×F = 4×F Subtract 2×F from each side of the above equation: C + 2×F - 2×F = 4×F - 2×F which makes C = 2×F
Hint #5
Substitute 2×F for E and C in eq.4: B + 2×F = A + 2×F Subtract 2×F from both sides of the equation above: B + 2×F - 2×F = A + 2×F - 2×F which makes B = A
Hint #6
Substitute 2×F for C, and A for B in eq.5: 2×F - A = A + A + F which becomes 2×F - A = 2×A + F In the above equation, add A to both sides, and subtract F from both sides: 2×F - A + A - F = 2×A + F + A - F which makes F = 3×A Divide both sides by 3: F ÷ 3 = 3×A ÷ 3 which makes ⅓×F = A and also makes ⅓×F = A = B
Solution
Substitute ⅓×F for A and B, 2×F for C and E, and 3×F for D in eq.1: ⅓×F + ⅓×F + 2×F + 3×F + 2×F + F = 26 which simplifies to 8⅔×F = 26 Divide both sides of the above equation by 8⅔: 8⅔×F ÷ 8⅔ = 26 ÷ 8⅔ which means F = 3 making A = B = ⅓×F = ⅓ × 3 = 1 C = E = 2×F = 2 × 3 = 6 D = 3×F = 3 × 3 = 9 and ABCDEF = 116963