Puzzle for April 4, 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.
* BC and DE are 2-digit numbers (not B×C or D×E).
Scratchpad
Help Area
Hint #1
Add F to both sides of eq.3: B + C + F = E - F + F which becomes eq.3a) B + C + F = E
Hint #2
In eq.4, replace E with B + C + F (from eq.3a): B + C + F - D = C + F Subtract both C and F from each side of the above equation: B + C + F - D - C - F = C + F - C - F which becomes B - D = 0 Add D to both sides: B - D + D = 0 + D which makes B = D
Hint #3
In eq.2, replace D with B: F - B = A + B Add B to each side of the above equation: F - B + B = A + B + B which becomes eq.2a) F = A + 2×B
Hint #4
In eq.5, substitute A + 2×B for F (from eq.2a): A + C = B + A + 2×B Subtract A from both sides: A + C - A = B + A + 2×B - A which makes C = 3×B
Hint #5
eq.6 may be written as: 10×D + E - A = A + 10×B + C Substitute B for D, and 3×B for C in the equation above: 10×B + E - A = A + 10×B + 3×B Add (A - 10×B) to both sides: 10×B + E - A + (A - 10×B) = A + 10×B + 3×B + (A - 10×B) which simplifies to eq.6a) E = 2×A + 3×B
Hint #6
Substitute 3×B for C, 2×A + 3×B for E (from eq.6a), and (A + 2×B) for F (from eq.2a) in eq.3: B + 3×B = 2×A + 3×B - (A + 2×B) which becomes 4×B = 2×A + 3×B - A - 2×B which becomes 4×B = A + B Subtract B from both sides: 4×B - B = A + B - B which makes 3×B = A
Hint #7
Substitute 3×B for A in eq.2a: F = 3×B + 2×B which makes F = 5×B
Hint #8
Substitute 3×B for A in eq.6a: E = 2×3×B + 3×B which makes E = 6×B + 3×B = 9×B
Solution
Substitute 3×B for A and C, B for D, 9×B for E, and 5×B for F in eq.1: 3×B + B + 3×B + B + 9×B + 5×B = 22 which simplifies to 22×B = 22 Divide both sides by 22: 22×B ÷ 22 = 22 ÷ 22 which makes B = 1 making A = C = 3×B = 3 × 1 = 3 D = B = 1 E = 9×B = 9 × 1 = 9 F = 5×B = 5 × 1 = 5 and ABCDEF = 313195