Puzzle for December 2, 2023 ( )
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.
* DE is a 2-digit number (not D×E).
Scratchpad
Help Area
Hint #1
Add C to both sides of eq.4: D + E + C = B - C + C which becomes eq.4a) D + E + C = B
Hint #2
In eq.3, replace B with D + E + C (from eq.4a): D + E + C + C = A + D which becomes D + E + 2×C = A + D Subtract D from each side of the above equation: D + E + 2×C - D = A + D - D which becomes eq.3a) E + 2×C = A
Hint #3
In eq.5, replace A with E + 2×C (from eq.3a): F - E = E + 2×C + C + E which becomes F - E = 2×E + 3×C Add E to both sides of the above equation: F - E + E = 2×E + 3×C + E which becomes eq.5a) F = 3×E + 3×C
Hint #4
In eq.2, substitute 3×E + 3×C for F (from eq.5a): 3×E + 3×C - B = B + D Add B to both sides of the equation above: 3×E + 3×C - B + B = B + D + B which becomes eq.2a) 3×E + 3×C = 2×B + D
Hint #5
Subtract D from both sides of eq.4a: D + E + C - D = B - D which becomea E + C = B - D Multiply both sides of the above equation by 3: 3 × (E + C) = 3 × (B - D) which becomes eq.4b) 3×E + 3×C = 3×B - 3×D
Hint #6
Substitute 3×B - 3×D for 3×E + 3×C (from eq.4b) into eq.2a: 3×B - 3×D = 2×B + D In the above equation, add 3×D to both sides, and subtract 2×B from both sides: 3×B - 3×D + 3×D - 2×B = 2×B + D + 3×D - 2×B which simplifies to B = 4×D
Hint #7
Substitute 4×D for B in eq.2: F - 4×D = 4×D + D which becomes F - 4×D = 5×D Add 4×D to both sides of the equation above: F - 4×D + 4×D = 5×D + 4×D which makes F = 9×D
Hint #8
eq.6 may be written as: 10×D + E = A - C + F Substitute 9×D for F in the above equation: 10×D + E = A - C + 9×D Subtract 9×D from both sides: 10×D + E - 9×D = A - C + 9×D - 9×D which becomes eq.6a) D + E = A - C
Hint #9
Substitute B - C for D + E (from eq.4) into eq.6a: B - C = A - C Add C to both sides of the equation above: B - C + C = A - C + C which makes B = A making B = A = 4×D
Hint #10
Substitute A for B in eq.3: A + C = A + D Subtract A from each side of the equation above: A + C - A = A - D - A which makes C = D
Hint #11
Substitute D for C, and 4×D for A in eq.3a: E + 2×D = 4×D Subtract 2×D from each side of the above equation: E + 2×D - 2×D = 4×D - 2×D which makes E = 2×D
Solution
Substitute 4×D for A and B, D for C, 2×D for E, and 9×D for F in eq.1: 4×D + 4×D + D + D + 2×D + 9×D = 21 which simplifies to 21×D = 21 Divide both sides of the above equation by 21: 21×D ÷ 21 = 21 ÷ 21 which means D = 1 making A = B = 4×D = 4 × 1 = 4 C = D = 1 E = 2×D = 2 × 1 = 2 F = 9×D = 9 × 1 = 9 and ABCDEF = 441129