Puzzle for February 2, 2021 ( )
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
In eq.2, replace D + E with F (from eq.6): B + F = C + F Subtract F from each side of the above equation: B + F – F = C + F – F which makes B = C
Hint #2
Add D to both sides of eq.4: D + E + F + D = B + C – D + D which becomes 2×D + E + F = B + C In the above equation, replace C with B: 2×D + E + F = B + B which becomes eq.4a) 2×D + E + F = 2×B
Hint #3
Add A, D, and F to both sides of eq.5: A – D – F + A + D + F = D – A + E + A + D + F which becomes 2×A = 2×D + E + F In the equation above, substitute 2×B for 2×D + E + F (from eq.4a): 2×A = 2×B Divide both sides by 2: 2×A ÷ 2 = 2×B ÷ 2 which makes A = B and also makes A = B = C
Hint #4
Substitute A for C in eq.3: A + E = A + D Subtract A from each side of the above equation: A + E – A = A + D – A which makes E = D
Hint #5
Substitute D for E in eq.6: F = D + D which makes F = 2×D
Hint #6
Substitute D for E, and 2×D for F in eq.4a: 2×D + D + 2×D = 2×B which becomes 5×D = 2×B Divide both sides of the above equation: 5×D ÷ 2 = 2×B ÷ 2 which makes 2½×D = B and also makes 2½×D = A = B = C
Solution
Substitute 2½×D for A and B and C, D for E, and 2×D for F in eq.1: 2½×D + 2½×D + 2½×D + D + D + 2×D = 23 which simplifies to 11½×D = 23 Divide both sides of the above equation by 11½: 11½×D ÷ 11½ = 23 ÷ 11½ which means D = 2 making A = B = C = 2½×D = 2½ × 2 = 5 E = D = 2 F = 2×D = 2 × 2 = 4 and ABCDEF = 555224