Puzzle for December 23, 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.2, replace E with C – B (from eq.5): B + C – B = A which becomes C = A
Hint #2
In eq.4, replace C with D – E (from eq.3): D + E = A + D – E In the above equation, subtract D from both sides, and add E to both sides: D + E – D + E = A + D – E – D + E which makes 2×E = A and also makes C = A = 2×E
Hint #3
In eq.2, substitute 2×E for A: B + E = 2×E Subtract E from both sides of the equation above: B + E – E = 2×E – E which makes B = E
Hint #4
Substitute 2×E for C in eq.3: 2×E = D – E Add E to both sides of the above equation: 2×E + E = D – E + E which makes 3×E = D
Hint #5
Substitute 2×E for A and C, and 3×E for D in eq.6: 2×E + 2×E = 3×E + E + F which becomes 4×E = 4×E + F Subtract 4×E from each side of the equation above: 4×E – 4×E = 4×E + F – 4×E which makes 0 = F
Solution
Substitute 2×E for A and C, 3×E for D, E for B, and 0 for F in eq.1: 2×E + E + 2×E + 3×E + E + 0 = 27 which simplifies to 9×E = 27 Divide both sides of the equation above by 9: 9×E ÷ 9 = 27 ÷ 9 which means E = 3 making A = C = 2×E = 2×3 = 6 B = E = 3 D = 3×E = 3×3 = 9 and ABCDEF = 636930