Puzzle for August 10, 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.4, substitute (D + E) for B (from eq.2): A – (D + E) = E – D which becomes A – D – E = E – D Add D and E to both sides of the above equation: A – D – E + D + E = E – D + D + E which simplifies to A = 2×E
Hint #2
Subtract D from both sides eq.5: C + F – D = A + B + D – D which becomes eq.5a) C + F – D = A + B
Hint #3
In eq.6, replace A + B with C + F – D (from eq.5a), and replace A with 2×E: C + D + E – 2×E = C + F – D – D – E which becomes C + D – E = C + F – 2×D – E In the above equation, subtract C from both sides, and add 2×D and E to both sides: C + D – E – C + 2×D + E = C + F – 2×D – E – C + 2×D + E which simplifies to eq.6a) 3×D = F
Hint #4
In eq.3, substitute 3×D for F (from eq.6a), 2×E for A, and (D + E) for B (from eq.2): 3×D – 2×E = 2×E – (D + E) which becomes 3×D – 2×E = 2×E – D – E which becomes 3×D – 2×E = E – D Add 2×E and D to both sides of the above equation: 3×D – 2×E + 2×E + D = E – D + 2×E + D which makes 4×D = 3×E Divide both sides by 4: 4×D ÷ 4 = 3×E ÷ 4 which makes D = ¾×E
Hint #5
Substitute (¾×E) for D in eq.6a: 3×(¾×E) = F which makes 2¼×E = F
Hint #6
Substitute ¾×E for D in eq.2: B = ¾×E + E which makes B = 1¾×E
Hint #7
Substitute 2¼×E for F, 2×E for A, 1¾×E for B, and ¾×E for D in eq.5: C + 2¼×E = 2×E + 1¾×E + ¾×E which becomes C + 2¼×E = 4½×E Subtract 2¼×E from each side of the equation above: C + 2¼×E – 2¼×E = 4½×E – 2¼×E which makes C = 2¼×E
Solution
Substitute 2×E for A, 1¾×E for B, 2¼×E for C and F, and ¾×E for D in eq.1: 2×E + 1¾×E + 2¼×E + ¾×E + E + 2¼×E = 40 which simplifies to 10×E = 40 Divide both sides of the above equation by 10: 10×E ÷ 10 = 40 ÷ 10 which means E = 4 making A = 2×E = 2 × 4 = 8 B = 1¾×E = 1¾ × 4 = 7 C = F = 2¼×E = 2¼ × 4 = 9 D = ¾×E = ¾ × 4 = 3 and ABCDEF = 879349