Puzzle for January 29, 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.
Scratchpad
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Hint #1
Add A + C to both sides of eq.2: B - A + A + C = A - C + A + C which becomes eq.2a) B + C = 2×A Add (A + E + F) to each side of eq.5: A - E - F + (A + E + F) = C - A - D + (A + E + F) which becomes eq.5a) 2×A = C - D + E + F
Hint #2
In eq.5a, replace 2×A with B + C (from eq.2a): B + C = C - D + E + F Add (D - C) to each side of the equation above: B + C + (D - C) = C - D + E + F + (D - C) which simplifies to B + D = E + F In the above equation, replace E + F with B (from eq.3): B + D = B Subtract B from each side: B + D - B = B - B which means D = 0
Hint #3
In eq.6, replace D with 0: C - E = 0 + E Add E to each side: C - E + E = 0 + E + E which means C = 2×E
Hint #4
Substitute 0 for D, and 2×E for C in eq.4: 2×E + 0 = F which means 2×E = F
Hint #5
In eq.3, substitute 2×E for F: B = E + 2×E which makes B = 3×E
Hint #6
Substitute 3×E for B, and 2×E for C in eq.2a: 3×E + 2×E = 2×A which becomes 5×E = 2×A Divide both sides of the above equation by 2: 5×E ÷ 2 = 2×A ÷ 2 which means 2½×E = A
Solution
Substitute 2½×E for A, 3×E for B, 2×E for C and F, and 0 for D in eq.1: 2½×E + 3×E + 2×E + 0 + E + 2×E = 21 which becomes 10½×E = 21 Divide both sides by 10½: 10½×E ÷ 10½ = 21 ÷ 10½ which means E = 2 making A = 2½×E = 2½ × 2 = 5 B = 3×E = 3 × 2 = 6 C = F = 2×E = 2 × 2 = 4 and ABCDEF = 564024