Puzzle for February 22, 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.
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Hint #1
Add B and C to both sides of eq.5: C - B + B + C = D - C + B + C which becomes 2×C = D + B which may be written as eq.5a) 2×C = B + D
Hint #2
Add D to both sides of eq.3: B + E + D = C - D + F + D which becomes B + E + D = C + F which may be written as eq.3a) B + D + E = C + F
Hint #3
In eq.3a, replace B + D with 2×C (from eq.5a): 2×C + E = C + F Subtract C from each side of the equation above: 2×C + E - C = C + F - C which becomes eq.3b) C + E = F
Hint #4
In eq.4, replace C + E with F (from eq.3b): D + F = F Subtract F from each side of the above equation: D + F - F = F - F which makes D = 0
Hint #5
In eq.5a, replace D with 0: 2×C = B + 0 which becomes 2×C = B
Hint #6
Substitute 0 for D, and (C + E) for F (from eq.3b) in eq.2: C - E = A - 0 - (C + E) which becomes C - E = A - C - E Add C and E to both sides of the above equation: C - E + C + E = A - C - E + C + E which makes 2×C = A
Hint #7
eq.6 may be written as: E = (A + B + D + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × E = 4 × (A + B + D + F) ÷ 4 which becomes eq.6a) 4×E = A + B + D + F
Hint #8
Substitute 2×C for A and B, 0 for D, and C + E for F (from eq.3b) into eq.6a: 4×E = 2×C + 2×C + 0 + C + E which becomes 4×E = 5×C + E Subtract E from both sides of the above equation: 4×E - E = 5×C + E - E which makes 3×E = 5×C Divide both sides by 3: 3×E ÷ 3 = 5×C ÷ 3 which makes E = 1⅔×C
Hint #9
Substitute 1⅔×C for E in eq.3b: C + 1⅔×C = F which makes 2⅔×C = F
Solution
Substitute 2×C for A and B, 0 for D, 1⅔×C for E, and 2⅔×C for F in eq.1: 2×C + 2×C + C + 0 + 1⅔×C + 2⅔×C = 28 which simplifies to 9⅓×C = 28 Divide both sides of the above equation by 9⅓: 9⅓×C ÷ 9⅓ = 28 ÷ 9⅓ which means C = 3 making A = B = 2×C = 2 × 3 = 6 E = 1⅔×C = 1⅔ × 3 = 5 F = 2⅔×C = 2⅔ × 3 = 8 and ABCDEF = 663058