Puzzle for November 24, 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 the left and right sides of eq.4 to the left and right sides of eq.3, respectively: C + F + B + E = A + E + A + F which becomes C + F + B + E = 2×A + E + F Subtract E and F from each side of the equation above: C + F + B + E - E - F = 2×A + E + F - E - F which becomes eq.3a) C + B = 2×A
Hint #2
Subtract the left and right sides of eq.3 from the left and right sides of eq.2, respectively: B + F - (C + F) = C + E - (A + E) which becomes B + F - C - F = C + E - A - E which becomes B - C = C - A Add C to both sides of the above equation: B - C + C = C - A + C which becomes eq.2a) B = 2×C - A
Hint #3
In eq.3a, replace B with 2×C - A (from eq.2a): C + 2×C - A = 2×A which becomes 3×C - A = 2×A Add A to both sides of the equation above: 3×C - A + A = 2×A + A which becomes 3×C = 3×A Divide both sides by 3: 3×C ÷ 3 = 3×A ÷ 3 which makes C = A
Hint #4
In eq.2a, substitute C for A: B = 2×C - C which makes B = C
Hint #5
Substitute A for C in eq.3: A + F = A + E Subtract A from both sides of the equation above: A + F - A = A + E - A which makes F = E
Hint #6
Substitute A + B + D - F for E + F (from eq.5) into eq.6: A + B + D - F = A + B + C - D - E Subtract A and B from each side of the above equation: A + B + D - F - A - B = A + B + C - D - E - A - B which becomes D - F = C - D - E Add F and D to both sides: D - F + F + D = C - D - E + F + D which becomes eq.6a) 2×D = C - E + F
Hint #7
Substitute E for F in eq.6a: 2×D = C - E + E which makes 2×D = C and also makes 2×D = B = C = A
Hint #8
Substitute E for F, and 2×D for A and B in eq.5: E + E = 2×D + 2×D + D - E which becomes 2×E = 5×D - E Add E to both sides of the equation above: 2×E + E = 5×D - E + E which makes 3×E = 5×D Divide both sides by 3: 3×E ÷ 3 = 5×D ÷ 3 which makes E = 1⅔×D and also makes E = F = 1⅔×D
Solution
Substitute 2×D for A and B and C, and 1⅔×D for E and F in eq.1: 2×D + 2×D + 2×D + D + 1⅔×D + 1⅔×D = 31 which simplifies to 10⅓×D = 31 Divide both sides of the above equation by 10⅓: 10⅓×D ÷ 10⅓ = 31 ÷ 10⅓ which means D = 3 making A = B = C = 2×D = 2 × 3 = 6 E = F = 1⅔×D = 1⅔ × 3 = 5 and ABCDEF = 666355