Puzzle for February 12, 2020 ( )
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 the left and right sides of eq.2 to the left and right sides of eq.3, respectively: A + C + B + C = D + E + F which becomes eq.3a) A + B + 2×C = D + E + F
Hint #2
In eq.3a, replace D + E with A + B – F (from eq.5): A + B + 2×C = A + B – F + F which becomes A + B + 2×C = A + B Subtract A and B from each side of the equation above: A + B + 2×C – A – B = A + B – A – B which becomes 2×C = 0 which means C = 0
Hint #3
In eq.3, replace C with 0: A + 0 = D which makes A = D
Hint #4
In eq.4, substitute A for D: E – F = A – A which becomes E – F = 0 Add F to each side of the equation above: E – F + F = 0 + F which makes E = F
Hint #5
Substitute 0 for C, and E for F in eq.2: B + 0 = E + E which makes B = 2×E
Hint #6
Substitute 2×E for B, 0 for C, A for D, and E for F in eq.6: A – 2×E – 0 + A – E = 2×E + 0 + E + E which becomes 2×A – 3×E = 4×E Add 3×E to both sides of the equation above: 2×A – 3×E + 3×E = 4×E + 3×E which makes 2×A = 7×E Divide both sides by 2: 2×A ÷ 2 = 7×E ÷ 2 which makes A = 3½×E and also makes D = A = 3½×E
Solution
Substitute 3½×E for A and D, 2×E for B, 0 for C, and E for F in eq.1: 3½×E + 2×E + 0 + 3½×E + E + E = 22 which simplifies to 11×E = 22 Divide both sides of the above equation by 11: 11×E ÷ 11 = 22 ÷ 11 which means E = 2 making A = D = 3½×E = 3½ × 2 = 7 B = 2×E = 2 × 2 = 4 F = E = 2 and ABCDEF = 740722