Puzzle for December 31, 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
Add A and F to both sides of eq.2: E - A + A + F = A - F + A + F which becomes eq.2a) E + F = 2×A eq.5 may be re-written as: eq.5a) F - B = A - (E + F)
Hint #2
In eq.5a, replace E + F with 2×A (from eq.2a): F - B = A - (2×A) which becomes F - B = -A Add B and A to both sides of the above equation: F - B + B + A = -A + B + A which becomes eq.5b) F + A = B
Hint #3
In eq.3, replace B with F + A (from eq.5b): F + A + D = C + F Subtract F from each side of the equation above: F + A + D - F = C + F - F which becomes eq.3a) A + D = C
Hint #4
eq.4 may be re-written as: E + F = A + D + C In the equation above, substitute C for A + D (from eq.3a): E + F = C + C which becomes eq.4a) E + F = 2×C
Hint #5
Substitute 2×A for E + F (from eq.2a) into eq.4a: 2×A = 2×C Divide both sides of the above equation by 2: 2×A ÷ 2 = 2×C ÷ 2 which makes A = C
Hint #6
Substitute A for C in eq.3a: A + D = A Subtract A from each side of the above equation: A + D - A = A - A which makes D = 0
Hint #7
Subtract F from both sides of eq.2a: E + F - F = 2×A - F which becomes eq.2b) E = 2×A - F
Hint #8
Substitute F + A for B (from eq.5b), 0 for D, and 2×A - F for E (from eq.2b) in eq.6: F = (F + A + 0 + 2×A - F) ÷ A which becomes F = (3×A) ÷ A which makes F = 3
Hint #9
Substitute 3 for F in eq.5b: eq.5c) 3 + A = B
Hint #10
Substitute 3 for F in eq.2b: eq.2c) E = 2×A - 3
Hint #11
Substitute A + 3 for B (from eq.5c), A for C, 0 for D, 2×A - 3 for E (from eq.2c), and 3 for F in eq.1: A + A + 3 + A + 0 + 2×A - 3 + 3 = 28 which simplifies to 5×A + 3 = 28 Subtract 3 from both sides of the above equation: 5×A + 3 - 3 = 28 - 3 which makes 5×A = 25 Divide both sides by 5: 5×A ÷ 5 = 25 ÷ 5 which makes A = 5 and also makes A = C = 5
Solution
Since A = 5, then: B = 3 + A = 3 + 5 = 8 (from eq.5c) E = 2×A - 3 = 2×5 - 3 = 10 - 3 = 7 (from eq.2c) and ABCDEF = 585073