Puzzle for January 3, 2023 ( )
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.4, replace B with E + F (from eq.3): E + F + E = A + F which becomes 2×E + F = A + F Subtract F from each side of the equation above: 2×E + F - F = A + F - F which makes 2×E = A
Hint #2
In eq.2, replace A with 2×E: 2×E = C + E Subtract E from both sides of the equation above: 2×E - E = C + E - E which makes E = C
Hint #3
In eq.5, substitute (E + F) for B (from eq.3), and 2×E for A: (E + F) + F = 2×E + D - (E + F) which becomes E + 2×F = 2×E + D - E - F which becomes E + 2×F = E + D - F In the above equation, subtract E from both sides, and add F to both sides: E + 2×F - E + F = E + D - F - E + F which simplifies to eq.5a) 3×F = D
Hint #4
Substitute 3×F for D, E + F for B (from eq.3), and E for C In eq.6: 3×F + F = E + F + E + E - F which becomes 4×F = 3×E Divide both sides of the above equation by 4: 4×F ÷ 4 = 3×E ÷ 4 which makes ¾×E = F
Hint #5
Substitute (¾×E) for F in eq.5a: 3×(¾×E) = D which makes 2¼×E = D
Hint #6
Substitute ¾×E for F in eq.3: B = E + ¾×E which makes B = 1¾×E
Solution
Substitute 2×E for A, 1¾×E for B, E for C, 2¼×E for D, and ¾×E for F in eq.1: 2×E + 1¾×E + E + 2¼×E + E + ¾×E = 35 which simplifies to 8¾×E = 35 Divide both sides of the above equation by 8¾: 8¾×E ÷ 8¾ = 35 ÷ 8¾ which means E = 4 making A = 2×E = 2 × 4 = 8 B = 1¾×E = 1¾ × 4 = 7 C = E = 4 D = 2¼×E = 2¼ × 4 = 9 F = ¾×E = ¾ × 4 = 3 and ABCDEF = 874943