Puzzle for November 7, 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.4, replace C with D + E (from eq.3): E - D = D + E - E 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 #2
In eq.3, replace E with 2×D: C = D + 2×D which makes C = 3×D
Hint #3
Add A and F to both sides of eq.5: F - A + A + F = B - F + A + F which becomes 2×F = B + A which is the same as eq.5a) 2×F = A + B
Hint #4
In eq.6, substitute 3×D for C, 2×D for E, and 2×F for A + B (from eq.5a): 3×D + 2×D = 2×F + D + F which becomes 5×D = 3×F + D Subtract D from both sides of the equation above: 5×D - D = 3×F + D - D which makes 4×D = 3×F Divide both sides by 3: 4×D ÷ 3 = 3×F ÷ 3 which makes 1⅓×D = F
Hint #5
Substitute 1⅓×D for F in eq.2: B = D + 1⅓×D which makes B = 2⅓×D
Hint #6
Substitute (1⅓×D) for F, and 2⅓×D for B in eq.5a: 2×(1⅓×D) = A + 2⅓×D which becomes 2⅔×D = A + 2⅓×D Subtract 2⅓×D from each side of the equation above: 2⅔×D - 2⅓×D = A + 2⅓×D - 2⅓×D which makes ⅓×D = A
Solution
Substitute ⅓×D for A, 2⅓×D for B, 3×D for C, 2×D for E, and 1⅓×D for F in eq.1: ⅓×D + 2⅓×D + 3×D + D + 2×D + 1⅓×D = 30 which simplifies to 10×D = 30 Divide both sides of the above equation by 10: 10×D ÷ 10 = 30 ÷ 10 which means D = 3 making A = ⅓×D = ⅓ × 3 = 1 B = 2⅓×D = 2⅓ × 3 = 7 C = 3×D = 3 × 3 = 9 E = 2×D = 2 × 3 = 6 F = 1⅓×D = 1⅓ × 3 = 4 and ABCDEF = 179364