Puzzle for February 19, 2019 ( )
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.5, replace C with B + E (from eq.2): A + D + E = B + E + F Subtract E from each side of the above equation: A + D + E - E = B + E + F - E which becomes A + D = B + F In eq.4, replace B + F with A + D: C + D = A + D Subtract D from both sides: C + D - D = A + D - D which makes C = A
Hint #2
In eq.2, substitute A for C: B + E = A In eq.3, substitute B + E for A: B + E = B + D Subtract B from each side: B + E - B = B + D - B which makes E = D
Hint #3
Substitute A for C, and D for E in eq.5: A + D + D = A + F Subtract A from each side of the above equation: A + D + D - A = A + F - A which simplifies to eq.5a) 2×D = F
Hint #4
Substitute A for C, D for E, and 2×D for F (from eq.5a) in eq.6: A + B + A = D + D + 2×D which becomes 2×A + B = 4×D In the above equation, substitute (B + D) for A (from eq.3): 2×(B + D) + B = 4×D which may be written as 2×B + 2×D + B = 4×D Subtract 2×D from both sides: 2×B + 2×D + B - 2×D = 4×D - 2×D which becomes 3×B = 2×D Divide each side by 2: 3×B ÷ 2 = 2×D ÷ 2 which makes 1½×B = D which also makes E = D = 1½×B
Hint #5
In eq.5a, replace D with 1½×B: 2×1½×B = F which means 3×B = F
Hint #6
In eq.2, replace E with 1½×B: B + 1½×B = C which means 2½×B = C which also means A = C = 2½×B
Solution
Substitute 2½×B for A and C, 1½×B for D and E, and 3×B for F in eq.1: 2½×B + B + 2½×B + 1½×B + 1½×B + 3×B = 24 which simplifies to 12×B = 24 Divide both sides of the above equation by 12: 12×B ÷ 12 = 24 ÷ 12 which makes B = 2 making A = C = 2½×B = 2½ × 2 = 5 D = E = 1½×B = 1½ × 2 = 3 F = 3×B = 3 × 2 = 6 and ABCDEF = 525336