Puzzle for June 22, 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
eq.5 may be re-written as: E = (A + B + D) ÷ 3 Multiply both sides of the above equation by 3: 3 × E = 3 × (A + B + D) ÷ 3 which becomes eq.5a) 3×E = A + B + D
Hint #2
Add D to both sides of eq.3: A + B + D = C - D + E + D which becomes eq.3a) A + B + D = C + E
Hint #3
In eq.5a, replace A + B + D with C + E (from eq.3a): 3×E = C + E Subtract E from each side of the equation above: 3×E - E = C + E - E which becomes 2×E = C
Hint #4
In eq.2, replace C with 2×E: F - E = 2×E - F Add E and F to both sides of the equation above: F - E + E + F = 2×E - F + E + F which makes 2×F = 3×E Divide both sides by 2: 2×F ÷ 2 = 3×E ÷ 2 which makes F = 1½×E
Hint #5
In eq.4, substitute A + B + D for C + E, and 1½×E for F (from eq.3a): B + A + B + D = A + D + 1½×E which becomes 2×B + A + D = A + D + 1½×E Subtract A and D from each side of the above equation: A + D + 1½×E - A - D = 2×B + A + D - A - D which simplifies to 1½×E = 2×B Divide both sides by 2: 1½×E ÷ 2 = 2×B ÷ 2 which makes ¾×E = B
Hint #6
eq.6 may be re-written as: F = (A + C + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × F = 3 × (A + C + E) ÷ 3 which becomes eq.6a) 3×F = A + C + E
Hint #7
Substitute 1½×E for F, and 2×E for C in eq.6a: 3×(1½×E) = A + 2×E + E which becomes 4½×E = A + 3×E Subtract 3×E from both sides of the equation above: 4½×E - 3×E = A + 3×E - 3×E which becomes 1½×E = A
Hint #8
Substitute 1½×E for A, and ¾×E for B in eq.5a: 3×E = 1½×E + ¾×E + D which becomes 3×E = 2¼×E + D Subtract 2¼×E each side of the above equation: 3×E - 2¼×E = 2¼×E + D - 2¼×E which makes ¾×E = D
Solution
Substitute 1½×E for A and F, ¾×E for B and D, and 2×E for C in eq.1: 1½×E + ¾×E + 2×E + ¾×E + E + 1½×E = 30 which simplifies to 7½×E = 30 Divide both sides of the above equation by 7½: 7½×E ÷ 7½ = 30 ÷ 7½ which means E = 4 making A = F = 1½×E = 1½ × 4 = 6 B = D = ¾×E = ¾ × 4 = 3 C = 2×E = 2 × 4 = 8 and ABCDEF = 638346