Puzzle for December 26, 2022 ( )
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 F with B + C (from eq.2): B + D = B + C - D In the equation above, subtract B from both sides, and add D to both sides: B + D - B + D = B + C - D - B + D which makes eq.3a) 2×D = C
Hint #2
In eq.6, replace C with 2×D: 2×D - E = D + E In the equation above, add E to both sides, and subtract D from both sides: 2×D - E + E - D = D + E + E - D which becomes D = 2×E
Hint #3
In eq.3a, substitute (2×E) for D: 2×(2×E) = C which makes 4×E = C
Hint #4
Substitute 4×E for C in eq.4: A - 4×E = 4×E - E which becomes A - 4×E = 3×E Add 4×E to both sides of the above equation: A - 4×E + 4×E = 3×E + 4×E which makes A = 7×E
Hint #5
Substitute 2×E for D, and 7×E for A in eq.5: 2×E + F = 7×E - E which becomes 2×E + F = 6×E Subtract 2×E from each side of the equation above: 2×E + F - 2×E = 6×E - 2×E which makes F = 4×E
Hint #6
Substitute 4×E for F and C in eq.2: 4×E = B + 4×E Subtract 4×E from both sides of the above equation: 4×E - 4×E = B + 4×E - 4×E which makes 0 = B
Solution
Substitute 7×E for A, 0 for B, 4×E for C and F, and 2×E for D in eq.1: 7×E + 0 + 4×E + 2×E + E + 4×E = 18 which simplifies to 18×E = 18 Divide both sides of the above equation by 18: 18×E ÷ 18 = 18 ÷ 18 which means E = 1 making A = 7×E = 7 × 1 = 7 C = F = 4×E = 4 × 1 = 4 D = 2×E = 2 × 1 = 2 and ABCDEF = 704214