Puzzle for February 8, 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 EF are 2-digit numbers (not A×B or E×F).
Scratchpad
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Hint #1
In eq.3, replace A with B + E (from eq.2): B + E + F = D + E Subtract E from both sides of the above equation: B + E + F - E = D + E - E which reduces to eq.3a) B + F = D
Hint #2
In eq.5, replace D with B + F (from eq.3a): A + B + B + F = E + F Subtract F from both sides of the above equation: A + B + B + F - F = E + F - F which becomes eq.5a) A + 2×B = E
Hint #3
In eq.2, substitute A + 2×B for E (from eq.5a): B + A + 2×B = A Subtract A from each side of the equation above: B + A + 2×B - A = A - A which simplifies to 3×B = 0 which means B = 0
Hint #4
Substitute 0 for B in eq.2: 0 + E = A which makes E = A
Hint #5
Substitute A for E in eq.3: A + F = D + A Subtract A from each side: A + F - A = D + A - A which makes D = F
Hint #6
eq.6 can be re-written as: 10×A + B + C = 10×E + F - C Add C to both sides of the equation above: 10×A + B + C + C = 10×E + F - C + C which becomes 10×A + B + 2×C = 10×E + F Substitute 0 for B, and A for E: 10×A + 0 + 2×C = 10×A + F Subtract 10×A from both sides: 10×A + 0 + 2×C - 10×A = 10×A + F - 10×A which simplifies to 2×C = F which also means D = F = 2×C
Hint #7
Substitute 2×C for D, and A for E in eq.4: C + 2×C = A + A which makes 3×C = 2×A Divide both sides by 2: 3×C ÷ 2 = 2×A ÷ 2 which means 1½×C = A which also means E = A = 1½×C
Solution
Substitute 1½×C for A and E, 0 for B, and 2×C for D and F in eq.1: 1½×C + 0 + C + 2×C + 1½×C + 2×C = 32 which simplifies to 8×C = 32 Divide both sides by 8: 8×C ÷ 8 = 32 ÷ 8 which means C = 4 making A = E = 1½×C = 1½ × 4 = 6 D = F = 2×C = 2 × 4 = 8 and ABCDEF = 604868