Puzzle for January 1, 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
Subtract E from both sides of eq.3: F – E = B + E – E which becomes F – E = B In eq.5, replace B with F – E: E + F = A + F – E – F which becomes E + F = A – E Add E to both sides of the equation above: E + F + E = A – E + E which becomes eq.5a) 2×E + F = A
Hint #2
In eq.2, replace A with 2×E + F (from eq.5a): 2×E + F + D = C + E + F Subtract E and F from both sides of the above equation: 2×E + F + D – E – F = C + E + F – E – F which becomes eq.2a) E + D = C
Hint #3
In eq.4, substitute E + D for C (from eq.2a): E + D = D + F Subtract D from both sides of the above equation: E + D – D = D + F – D which makes E = F
Hint #4
Substitute E for F in eq.3: E = B + E Subtract E from both sides of the equation above: E – E = B + E – E which makes 0 = B
Hint #5
Substitute E for F in eq.5a: 2×E + E = A which makes 3×E = A
Hint #6
Substitute 3×E for A, (E + D) for C (from eq.2a), and E for F in eq.6: D – 3×E = 3×E – (E + D) + E – E which becomes D – 3×E = 3×E – E – D which becomes D – 3×E = 2×E – D Add D and 3×E to each side of the above equation: D – 3×E + D + 3×E = 2×E – D + D + 3×E which makes 2×D = 5×E Divide both sides by 2: 2×D ÷ 2 = 5×E ÷ 2 which makes D = 2½×E
Hint #7
Substitute 2½×E for D in eq.2a: E + 2½×E = C which makes 3½×E = C
Solution
Substitute 3×E for A, 0 for B, 3½×E for C, 2½×E for D, and E for F in eq.1: 3×E + 0 + 3½×E + 2½×E + E + E = 22 which simplifies to 11×E = 22 Divide both sides of the equation above by 11: 11×E ÷ 11 = 22 ÷ 11 which means E = 2 making A = 3×E = 3 × 2 = 6 C = 3½×E = 3½ × 2 = 7 D = 2½×E = 2½ × 2 = 5 F = E = 2 and ABCDEF = 607522