Puzzle for June 9, 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 positive integer.
* EF is a 2-digit number (not E×F).
Scratchpad
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Hint #1
Add D and C to both sides of eq.5: C - D + D + C = A + B - C + E + F + D + C which becomes eq.5a) 2×C = A + B + E + F + D eq.1 may be written as: eq.1a) A + B + E + F + D + C = 27
Hint #2
In eq.1a, replace A + B + E + F + D with 2×C (from eq.5a): 2×C + C = 27 which becomes 3×C = 27 Divide both sides of the above equation by 3: 3×C ÷ 3 = 27 ÷ 3 which makes C = 9
Hint #3
In eq.5a, replace C with 9: 2×9 = A + B + E + F + D which makes eq.5b) 18 = A + B + D + E + F
Hint #4
Subtract the left and right sides of eq.2 from the left and right sides of eq.3, respectively: E + F - (B + F) = A + B - (A + E) which becomes E + F - B - F = A + B - A - E which becomes E - B = B - E Add B and E to both sides of the above equation: E - B + B + E = B - E + B + E which becomes 2×E = 2×B Divide both sides of the above equation by 2: 2×E ÷ 2 = 2×B ÷ 2 which makes E = B
Hint #5
In eq.3, substitute B for E: B + F = A + B Subtract B from each side of the above equation: B + F - B = A + B - B which makes F = A
Hint #6
eq.4 may be re-written as: B - F = A - E Add the left and right sides of the above equation to the left and right sides of eq.2, respectively: B + F + B - F = A + E + A - E which becomes 2×B = 2×A Divide both sides of the above equation by 2: 2×B ÷ 2 = 2×A ÷ 2 which makes B = A and therefore makes E = B = A = F
Hint #7
eq.6 may be written as: B × C = 10×E + F - (A × D) Substitute 9 for C, and B for E and F and A in the above equation: B × 9 = 10×B + B - (B × D) which becomes eq.6a) 9×B = 11×B - (B × D)
Hint #8
In eq.6a, subtract 9×B from both sides, and add (B × D) to both sides: 9×B - 9×B + (B × D) = 11×B - (B × D) - 9×B + (B × D) which becomes B × D = 2×B Divide both sides of the above equation by B: B × D ÷ B = 2×B ÷ B which makes D = 2
Solution
Substitute B for A and E and F, and 2 for D in eq.5b: 18 = B + B + 2 + B + B which becomes 18 = 4×B + 2 Subtract 2 from each side of the equation above: 18 - 2 = 4×B + 2 - 2 which makes 16 = 4×B Divide both sides by 4: 16 ÷ 4 = 4×B ÷ 4 which makes 4 = B and also makes E = A = F = B = 4 making ABCDEF = 449244