Puzzle for March 14, 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 A, D, and E to both sides of eq.4: C - A + A + D + E = B - D - E + A + D + E which becomes eq.4a) C + D + E = B + A
Hint #2
In eq.5, replace C + D + E with B + A (from eq.4a): A + B + F = B + A Subtract A and B from both sides of the above equation: A + B + F - A - B = B + A - A - B which makes F = 0
Hint #3
In eq.1, replace F with 0: A = B - 0 which makes A = B
Hint #4
In eq.2, replace F with 0: C = E - 0 which makes C = E
Hint #5
In eq.6, substitute 0 for F, A for B, and C for E: D - 0 = (A ÷ A) + (C ÷ C) which becomes D = 1 + 1 which makes D = 2
Hint #6
Substitute 2 for D, and C for E in eq.3: B + 2 = C + C - 2 which becomes B + 2 = 2×C - 2 Subtract 2 from each side of the equation above: B + 2 - 2 = 2×C - 2 - 2 which makes B = 2×C - 4 and also makes eq.3a) A = B = 2×C - 4
Hint #7
Substitute 2 for D, C for E, and 2×C - 4 for B and A (from eq.3a) in eq.4a: C + 2 + C = 2×C - 4 + 2×C - 4 which becomes 2×C + 2 = 4×C - 8 In the above equation, subtract 2×C from both sides, and add 8 to both sides: 2×C + 2 - 2×C + 8 = 4×C - 8 - 2×C + 8 which simplifies to 10 = 2×C Divide both sides by 2: 10 ÷ 2 = 2×C ÷ 2 which makes 5 = C and also makes E = C = 5
Solution
Substitute 5 for C in eq.3a: A = B = 2×5 - 4 which becomes A = B = 10 - 4 which makes A = B = 6 and makes ABCDEF = 665250