Puzzle for March 23, 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.
Scratchpad
Help Area
Hint #1
In eq.5, replace D with B + C (from eq.2): B + E = C + B + C - B which becomes eq.5a) B + E = 2×C
Hint #2
Add C to both sides of eq.4: C - B + C = F - C + C which becomes eq.4a) 2×C - B = F In eq.4a, replace 2×C with B + E (from eq.5a): B + E - B = F which makes E = F
Hint #3
eq.6 may be written as: D = (A + B + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × D = 3 × (A + B + E) ÷ 3 which becomes eq.6a) 3×D = A + B + E
Hint #4
Subtract B from both sides of eq.3: F - B = A + B - B which becomes F - B = A In eq.6a, substitute F - B for A, and F for E: 3×D = F - B + B + F which makes eq.3a) 3×D = 2×F
Hint #5
Multiply both sides of eq.5 by 3: 3 × (B + E) = 3 × (C + D - B) which becomes 3×B + 3×E = 3×C + 3×D - 3×B In the above equation, substitute F for E, and 2×F for 3×D (from eq.3a): 3×B + 3×F = 3×C + 2×F - 3×B Subtract 2×F from both sides, and add 3×B to both sides: 3×B + 3×F - 2×F + 3×B = 3×C + 2×F - 3×B - 2×F + 3×B which simplifies to eq.5b) 6×B + F = 3×C
Hint #6
Substitute 2×C - B for F (from eq.4a) in eq.5b: 6×B + 2×C - B = 3×C which becomes 5×B + 2×C = 3×C Subtract 2×C from each side of the equation above: 5×B + 2×C - 2×C = 3×C - 2×C which makes 5×B = C
Hint #7
Substitute (5×B) for C in eq.4a: 2×(5×B) - B = F which becomes 10×B - B = F which makes 9×B = F and also makes 9×B = F = E
Hint #8
Substitute (9×B) for F in eq.3a: 3×D = 2×(9×B) which makes 3×D = 18×B Divide both sides of the above equation by 3: 3×D ÷ 3 = 18×B ÷ 3 which makes D = 6×B
Hint #9
Substitute 9×B for F in eq.3: 9×B = A + B Subtract B from each side of the equation above: 9×B - B = A + B - B which makes 8×B = A
Solution
Substitute 8×B for A, 5×B for C, 6×B for D, and 9×B for E and F in eq.1: 8×B + B + 5×B + 6×B + 9×B + 9×B = 38 which simplifies to 38×B = 38 Divide both sides of the above equation by 38: 38×B ÷ 38 = 38 ÷ 38 which means B = 1 making A = 8×B = 8 × 1 = 8 C = 5×B = 5 × 1 = 5 D = 6×B = 6 × 1 = 6 E = F = 9×B = 9 × 1 = 9 and ABCDEF = 815699