Puzzle for November 17, 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 positive integer.
Once again, our thanks go out to Judah S (age 16) for sending us another interesting puzzle! Thank you, Judah!
Scratchpad
Help Area
Hint #1
In eq.3, substitute (B + F) for A (from eq.1): C - D = D - (B + F) which becomes C - D = D - B - F Add D, B, and F to both sides of the above equation: C - D + D + B + F = D - B - F + D + B + F which becomes C + B + F = 2×D which may be written as eq.3a) B + C + F = 2×D
Hint #2
In eq.3a, replace C + F with D (from eq.2): B + D = 2×D Subtract D from each side of the equation above: B + D - D = 2×D - D which makes B = D
Hint #3
Add F to both sides of eq.4: E + F + F = B + C - F + F which becomes eq.4a) E + 2×F = B + C Add C and E to both sides of eq.5: E - C + C + E = B + D - E + C + E which becomes 2×E = B + D + C which may be written as eq.5a) 2×E = B + C + D
Hint #4
In eq.5a, replace B + C with E + 2×F (from eq.4a): 2×E = E + 2×F + D Subtract E from each side of the above equation: 2×E - E = E + 2×F + D - E which becomes eq.5b) E = 2×F + D
Hint #5
In eq.4a, replace E with 2×F + D (from eq.5b), and B with D: 2×F + D + 2×F = D + C which becomes 4×F + D = D + C Subtract D from both sides of the above equation: 4×F + D - D = D + C - D which makes 4×F = C
Hint #6
In eq.2, substitute 4×F for C: D = 4×F + F which makes D = 5×F and also makes B = D = 5×F
Hint #7
Substitute 5×F for D in eq.5b: E = 2×F + 5×F which makes E = 7×F
Hint #8
Substitute 5×F for B in eq.1: A = 5×F + F which makes A = 6×F
Hint #9
Substitute 6×F for A, 5×F for D and B, and 7×F for E in eq.6: 6×F × (5×F + F) = (5×F × 7×F) + F which becomes 6×F × 6×F = 35×F² + F which becomes 36×F² = 35×F² + F Subtract 35×F² from each side of the equation above: 36×F² - 35×F² = 35×F² + F - 35×F² which makes F² = F Divide both sides by F: F² ÷ F = F ÷ F which means F = 1
Solution
Since F = 1, then: A = 6×F = 6 × 1 = 6 B = D = 5×F = 5 × 1 = 5 C = 4×F = 4 × 1 = 4 E = 7×F = 7 × 1 = 7 and ABCDEF = 654571