Math Puzzles and Word Problems

This is a classic logic puzzle. If Alex is the knight (truth-teller), then Cody is the knave (liar), but Cody says he's the spy - which a knave cannot say truthfully. By systematic elimination, we find that Brook is the spy, Cody is the knave, and Alex is the knight.

The sequence M T W T represents the first four letters of days of the week: Monday, Tuesday, Wednesday, Thursday. The pattern continues with Friday (F), Saturday (S), and Sunday (S).

If 1.5 hens lay 1.5 eggs in 1.5 days, then 1.5 hens lay 1 egg per day (1.5 / 1.5). If 1.5 hens lay 1 egg per day, then 1 hen lays $1 / 1.5 = 2/3$ of an egg per day. Thus, the daily rate per hen is 2/3.

Let the number be $10x + y$. From the first condition, $10x + y = 4(x + y) + 3$, which simplifies to $2x - y = 1$. From the second, $10x + y = 3xy + 5$. Substituting $y = 2x - 1$ into the second equation gives $10x + 2x - 1 = 3x(2x - 1) + 5$, leading to $6x^2 - 15x + 6 = 0$. Solving gives $x=2$, so $y=3$. The number is 23.

To find the number of even factors, first express the number in prime form: $2^{14} \times 3^9 \times 5^6$. The total number of factors is $(14+1)(9+1)(6+1) = 1050$. The number of odd factors is $(9+1)(6+1) = 70$. Subtracting odd from total gives $1050 - 70 = 980$ even factors.

Using conservation of momentum: m1v1 + m2v2 = (m1+m2)vf. Himasur (500 kg) at 50 m/s and superheroes (200 kg) at v collide. After collision, combined system moves at 50 m/s into black hole: (500)(50) + (200)(-v) = (700)(50). Solving gives v = -200 m/s (200 m/s toward Himasur). Option B matches this result.

The pattern is 25 × n^3 ± 1 where n = 1, 2, 3, 4. Specifically: 25×1+1=26, 25×8-1=199, 25×27+1=676, 25×64-1=1594. The adjustment alternates between +1 and -1. Therefore the next term after 194 is 1594 (noting the question asks for the next term after 194).

The series follows 25 × n^3 ± 1 for n=1,2,3: 25×1+1=26, 25×8-1=199, 25×27+1=676. The multipliers are cubes (1, 8, 27 = 1³, 2³, 3³) and adjustments alternate (+1, -1, +1). Therefore the 4th term is 676, which is 26².

The differences between terms are: 5-1=4, 18-5=13, 67-18=49, 197-67=130. These differences (4, 13, 49, 130) follow a pattern where second differences are 3², 6², 9² (i.e., 9, 36, 81). Each second difference is a multiple of 3 squared: 9×1², 9×2², 9×3².

The pattern involves alternating operations: 11+1=12, 12-3=9, 9-17=-8. The absolute differences (1, 3, 17) follow the pattern 2^n ± 1 with alternating signs. More simply, this can be modeled as a cubic sequence f(n) = -5/3n³ + 8n² - 56/3n + 22 for n=1,2,3,4, giving terms 11, 12, 9, -8.

The pattern is based on increasing differences: 10-7=3, 21-10=11, 43-21=22. The differences (3, 11, 22) do not follow a simple linear progression that leads to 79. If the next number were 79, the difference would be 36. Without a clear consistent rule, the statement is False.

Son's age = 2. Father's age = 11 × 2 = 22. Grandfather's age = 2 × 22 = 44. The grandfather is 44 years old. Option A (2) is the son's age, B (22) is the father's age, and D (88) would require 4× multiplier.

Son's age = 2. Father's age = 11 × 2 = 22. Grandfather's age = 2 × 22 = 44. The grandfather is 44 years old. Option A (2) is the son's age, B (22) is the father's age, and D (88) would require 4× multiplier.