Puzzle for December 16, 2022 ( )
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 the left and right sides of both eq.2 and eq.3 to the left and right sides of eq.5, respectively: E - D + C + A = F - E + D + E + E + F which becomes E - D + C + A = 2×F + D + E In the equation above, subtract E from both sides, and add D to both sides: E - D + C + A - E + D = 2×F + D + E - E + D which becomes eq.5a) C + A = 2×F + 2×D
Hint #2
Add F and D to both sides of eq.4: D - F + F + D = A - D + F + D which becomes eq.4a) 2×D = A + F
Hint #3
In eq.5a, replace 2×D with A + F (from eq.4a): C + A = 2×F + A + F which becomes C + A = 3×F + A Subtract A from each side of the equation above: C + A - A = 3×F + A - A which becomes eq.5b) C = 3×F
Hint #4
In eq.2, replace C with 3×F: 3×F = D + E Subtract E from each side of the above equation: 3×F - E = D + E - E which becomes eq.2a) 3×F - E = D
Hint #5
In eq.5, substitute (3×F - E) for D (from eq.2a): E - (3×F - E) = F - E which becomes E - 3×F + E = F - E which becomes 2×E - 3×F = F - E Add 3×F and E to both sides of the above equation: 2×E - 3×F + 3×F + E = F - E + 3×F + E which makes 3×E = 4×F Divide both sides by 4: 3×E ÷ 4 = 4×F ÷ 4 which makes ¾×E = F
Hint #6
Substitute ¾×E for F in eq.5b: C = 3×(¾×E) which makes C = 2¼×E
Hint #7
Substitute ¾×E for F in eq.2a: 3×(¾×E) - E = D which becomes 2¼×E - E = D which makes 1¼×E = D
Hint #8
Substitute (1¼×E) for D, and ¾×E for F in eq.4a: 2×(1¼×E) = A + ¾×E which becomes 2½×E = A + ¾×E Subtract ¾×E from each side of the equation above: 2½×E - ¾×E = A + ¾×E - ¾×E which makes 1¾×E = A
Hint #9
Substitute 1¼×E for D, 2¼×E for C, and ¾×E for F in eq.6: B + 1¼×E = 2¼×E + ¾×E which becomes B + 1¼×E = 3×E Subtract 1¼×E from each side of the equation above: B + 1¼×E - 1¼×E = 3×E - 1¼×E which makes B = 1¾×E
Solution
Substitute 1¾×E for A and B, 2¼×E for C, 1¼×E for D, and ¾×E for F in eq.1: 1¾×E + 1¾×E + 2¼×E + 1¼×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 = B = 1¾×E = 1¾ × 4 = 7 C = 2¼×E = 2¼ × 4 = 9 D = 1¼×E = 1¼ × 4 = 5 F = ¾×E = ¾ × 4 = 3 and ABCDEF = 779543