Puzzle for October 2, 2020 ( )
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.
* DE is a 2-digit number (not D×E).
Scratchpad
Help Area
Hint #1
Add F to both sides of eq.3: F – A + F = D + E – F + F which becomes eq.3a) 2×F – A = D + E In eq.4, add C and F to both sides, and subtract A from both sides: A – D + E – F + C + F – A = F – C + C + F – A which becomes eq.4a) –D + E + C = 2×F – A
Hint #2
In eq.4a, replace 2×F – A with D + E (from eq.3a): –D + E + C = D + E In the equation above, add D to both sides, and subtract E from both sides: –D + E + C + D – E = D + E + D – E which becomes C = 2×D
Hint #3
In eq.2, replace C with 2×D: 2×D – E = E – D Add D and E to each side of the above equation: 2×D – E + D + E = E – D + D + E which makes 3×D = 2×E Divide both sides by 2: 3×D ÷ 2 = 2×E ÷ 2 which makes 1½×D = E
Hint #4
eq.5 may be written as: E × F = C + 10×D + E In the above equation, substitute 1½×D for E, and 2×D for C: 1½×D × F = 2×D + 10×D + 1½×D which becomes 1½×D × F = 13½×D Divide both sides by 1½×D: 1½×D × F ÷ 1½×D = 13½×D ÷ 1½×D which makes F = 9
Hint #5
eq.6 may be written as: 10×D + E + E – D = C × E which becomes 9×D + 2×E = C × E In the above equation, substitute (1½×D) for E, and 2×D for C: 9×D + 2×(1½×D) = 2×D × (1½×D) which becomes 9×D + 3×D = 2×1½×D×D which makes 12×D = 3×D² Divide both sides by 3×D: 12×D ÷ 3×D = 3×D² ÷ 3×D which makes 4 = D making C = 2×D = 2 × 4 = 8 E = 1½×D = 1½ × 4 = 6
Hint #6
Substitute 4 for D, 6 for E, 9 for F, and 8 for C in eq.4: eq.4) A – 4 + 6 – 9 = 9 – 8 which becomes A – 7 = 1 Add 7 to both sides of the above equation: A – 7 + 7 = 1 + 7 which makes A = 8
Solution
Substitute 6 for E, and 8 for A in eq.1: B + 6 = 8 Subtract 6 from both sides: B + 6 – 6 = 8 – 6 which makes B = 2 and makes ABCDEF = 828469