Puzzle for July 10, 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
In eq.2, replace F with B + E (from eq.4): E + B + E = A + B which becomes B + 2×E = A + B Subtract B from each side of the above equation: B + 2×E - B = A + B - B which makes 2×E = A
Hint #2
In eq.6, substitute (B + E) for F (from eq.4): C - E = B + D - (B + E) which is equivalent to C - E = B + D - B - E which becomes C - E = D - E Add E to both sides: C - E + E = D - E + E which makes C = D
Hint #3
Substitute C for D, 2×E for A, and B + E for F (from eq.4) into eq.3: B + C + C = 2×E + B + E which becomes B + 2×C = 3×E + B Subtract B from both sides of the above equation: B + 2×C - B = 3×E + B - B which means 2×C = 3×E Divide both sides by 2: 2×C ÷ 2 = 3×E ÷ 2 which makes C = 1½×E and which also makes D = C = 1½×E
Hint #4
Substitute 2×E for A, and 1½×E for C into eq.5: 2×E - F = B - 1½×E Add 1½×E to both sides of the above equation: 2×E - F + 1½×E = B - 1½×E + 1½×E which becomes eq.5a) 3½×E - F = B
Hint #5
Substitute 3½×E - F for B (from eq.5a) into eq.4: F = 3½×E - F + E which becomes F = 4½×E - F Add F to both sides: F + F = 4½×E - F + F which means 2×F = 4½×E Divide both sides by 2: 2×F ÷ 2 = 4½×E ÷ 2 which makes F = 2¼×E
Hint #6
Substitute 2¼×E for F into eq.5a: 3½×E - 2¼×E = B which makes 1¼×E = B
Solution
Substitute 2×E for A, 1¼×E for B, 1½×E for C and D, and 2¼×E for F in eq.1: 2×E + 1¼×E + 1½×E + 1½×E + E + 2¼×E = 38 which simplifies to 9½×E = 38 Divide both sides by 9½: 9½×E ÷ 9½ = 38 ÷ 9½ which means E = 4 making A = 2×E = 2 × 4 = 8 B = 1¼×E = 1¼ × 4 = 5 C = D = 1½×E = 1½ × 4 = 6 F = 2¼×E = 2¼ × 4 = 9 and ABCDEF = 856649