Puzzle for February 22, 2019 ( )
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.
* AB and CD are 2-digit numbers (not A×B or C×D).
Scratchpad
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Hint #1
Add C to both sides of eq.3: D + C = B - C + C which becomes D + C = B which may also be written as C + D = B In eq.5, replace C + D with B: eq.5a) B = E - F
Hint #2
Subtract the left and right sides of eq.5a from the left and right sides of eq.2, respectively: B + C - B = E + F - (E - F) which becomes C = E + F - E + F which becomes C = 2×F
Hint #3
Add (A - D) to each side of eq.4: E - A + (A - D) = A + D + (A - D) which becomes eq.4a) E - D = 2×A Add (F - D) to both sides of eq.5: C + D + (F - D) = E - F + (F - D) which becomes eq.5b) C + F = E - D
Hint #4
In eq.5b, substitute 2×F for C, and 2×A for E - D (from eq.4a): 2×F + F = 2×A which becomes 3×F = 2×A Divide both sides of the equation above by 2: 3×F ÷ 2 = 2×A ÷ 2 which makes 1½×F = A
Hint #5
In eq.4a, substitute (1½×F) for A: E - D = 2×(1½×F) which becomes E - D = 3×F Add (D - 3×F) to each side of the above equation: E - D + (D - 3×F) = 3×F + (D - 3×F) eq.4b) E - 3×F = D
Hint #6
eq.6 may be written as: 10×A + B = 10×C + D - E Substitute (1½×F) for A, and (2×F) for C in the above equation: 10×(1½×F) + B = 10×(2×F) + D - E which becomes 15×F + B = 20×F + D - E Subtract 15×F from both sides: 15×F + B - 15×F = 20×F + D - E - 15×F which becomes eq.6a) B = 5×F + D - E
Hint #7
In eq.6a, substitute E - F for B (from eq.5a), and E - 3×F for D (from eq.4b): E - F = 5×F + E - 3×F - E which becomes E - F = 2×F Add F to both sides: E - F + F = 2×F + F which makes E = 3×F
Hint #8
Substitute 3×F for E in eq.4b: 3×F - 3×F = D which means 0 = D
Hint #9
Substitute 0 for D, and 2×F for C in eq.3: 0 = B - 2×F Add 2×F to both sides: 0 + 2×F = B - 2×F + 2×F which makes 2×F = B
Solution
Substitute 1½×F for A, 2×F for B and C, 0 for D, and 3×F for E in eq.1: 1½×F + 2×F + 2×F + 0 + 3×F + F = 19 which simplifies to 9½×F = 19 Divide both sides by 9½: 9½×F ÷ 9½ = 19 ÷ 9½ which makes F = 2 making A = 1½×F = 1½ × 2 = 3 B = C = 2×F = 2 × 2 = 4 E = 3×F = 3 × 2 = 6 and ABCDEF = 344062