Puzzle for November 5, 2020 ( )
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.3, replace D with B + E (from eq.2): B + C = B + E + E which becomes B + C = B + 2×E Subtract B from each side of the equation above: B + C – B = B + 2×E – B which makes C = 2×E
Hint #2
In eq.4, replace C with 2×E: 2×E + E = D + F which becomes 3×E = D + F Subtract D from each side of the above equation: 3×E – D = D + F – D which makes eq.4a) 3×E – D = F
Hint #3
Add D and B to both sides of eq.5: D – B + D + B = A – D + D + B which becomes 2×D = A + B In eq.6, substitute 2×D for A + B, and 2×E for C: 2×D + D = 2×E + F which becomes eq.6a) 3×D = 2×E + F
Hint #4
Substitute 3×E – D for F (from eq.4a) in eq.6a: 3×D = 2×E + 3×E – D which becomes 3×D = 5×E – D Add D to both sides of the above equation: 3×D + D = 5×E – D + D which makes 4×D = 5×E Divide both sides by 4: 4×D ÷ 4 = 5×E ÷ 4 which makes D = 1¼×E
Hint #5
Substitute 1¼×E for D in eq.2: 1¼×E = B + E Subtract E from both sides of the equation above: 1¼×E – E = B + E – E which makes ¼×E = B
Hint #6
Substitute 1¼×E for D in eq.4a: 3×E – 1¼×E = F which makes 1¾×E = F
Hint #7
Substitute 1¼×E for D, and ¼×E for B in eq.5: 1¼×E – ¼×E = A – 1¼×E which becomes E = A – 1¼×E Add 1¼×E to both sides of the equation above: E + 1¼×E = A – 1¼×E + 1¼×E which makes 2¼×E = A
Solution
Substitute 2¼×E for A, ¼×E for B, 2×E for C, 1¼×E for D, and 1¾×E for F in eq.1: 2¼×E + ¼×E + 2×E + 1¼×E + E + 1¾×E = 34 which simplifies to 8½×E = 34 Divide both sides of the equation above by 8½: 34×E ÷ 8½ = 34 ÷ 8½ which means E = 4 making A = 2¼×E = 2¼ × 4 = 9 B = ¼×E = ¼ × 4 = 1 C = 2×E = 2 × 4 = 8 D = 1¼×E = 1¼ × 4 = 5 F = 1¾×E = 1¾ × 4 = 7 and ABCDEF = 918547