Puzzle for December 14, 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
Help Area
Hint #1
eq.5 may be written as: A - B + D = E + F - A Subtract the left and right sides of the above equation from the left and right sides of eq.6, respectively: A - B + C - (A - B + D) = E + F - C - D - (E + F - A) which becomes A - B + C - A + B - D = E + F - C - D - E - F + A which becomes C - D = -C - D + A Add D and C to both sides of the equation above: C - D + D + C = -C - D + A + D + C which makes 2×C = A
Hint #2
In eq.3, replace A with 2×C: F - C = 2×C - F Add C and F to both sides of the above equation: F - C + C + F = 2×C - F + C + F which makes 2×F = 3×C Divide both sides by 2: 2×F ÷ 2 = 3×C ÷ 2 which makes F = 1½×C
Hint #3
In eq.2, substitute 1½×C for F, and 2×C for A: B + 1½×C = 2×C - B In the above equation, subtract 1½×C from both sides, and add B to both sides: B + 1½×C - 1½×C + B = 2×C - B - 1½×C + B which becomes 2×B = ½×C Divide both sides by 2: 2×B ÷ 2 = ½×C ÷ 2 which makes B = ¼×C
Hint #4
Substitute 1½×C for F, and ¼×C for B in eq.4: C + 1½×C = ¼×C + E which becomes 2½×C = ¼×C + E Subtract ¼×C from each side of the equation above: 2½×C - ¼×C = ¼×C + E - ¼×C which makes 2¼×C = E
Hint #5
Substitute 2¼×C for E, 1½×C for F, 2×C for A, and ¼×C for B in eq.5: 2¼×C + 1½×C - 2×C = 2×C - ¼×C + D which becomes 1¾×C = 1¾×C + D Subtract 1¾×C from each side of the above equation: 1¾×C - 1¾×C = 1¾×C + D - 1¾×C which makes 0 = D
Solution
Substitute 2×C for A, ¼×C for B, 0 for D, 2¼×C and E, and 1½×C for F in eq.1: 2×C + ¼×C + C + 0 + 2¼×C + 1½×C = 28 which simplifies to 7×C = 28 Divide both sides of the above equation by 7: 7×C ÷ 7 = 28 ÷ 7 which means C = 4 making A = 2×C = 2 × 4 = 8 B = ¼×C = ¼ × 4 = 1 E = 2¼×C = 2¼ × 4 = 9 F = 1½×C = 1½ × 4 = 6 and ABCDEF = 814096