Mathematical Puzzles and Problem Solving

A standard deck has 4 suits. Removing Jack (11), Queen (12), King (13), and Ace (1) leaves cards 2 through 10. The sum of cards 2 to 10 in one suit is 54. For 4 suits, the total sum is $54 \times 4 = 216$.

Let $M, T, W, Th$ be the temperatures. $M+T+W = 111$ and $T+W+Th = 102$. Subtracting gives $M - Th = 9$. Since $Th = 4/5 M$, we have $M - 4/5 M = 9 \implies M/5 = 9 \implies M = 45$. Thus, $Th = 4/5 \times 45 = 36$ degrees.

A's speed is 50 km/hr. B is 50% faster, so B's speed is 75 km/hr. By 10 A.M., A has traveled 50 km, leaving 250 km between them. Traveling in opposite directions (towards each other), their relative speed is 125 km/hr. They meet in $250 / 125 = 2$ hours, which is 12:00 P.M.

Using a balance scale and a 1kg stone, you can divide the rice in 5 weighings: 1) Split 31kg + 1kg stone into 16kg and 16kg (15kg rice + stone). 2) Split 16kg into 8kg and 8kg. 3) Split 8kg into 4kg and 4kg. 4) Split 4kg into 2kg and 2kg. 5) Split 2kg into 1kg and 1kg.

The series is a tribonacci-like sequence where each term is the sum of the preceding three terms: $0+1+1=2$, $1+1+2=4$, $1+2+4=7$, $2+4+7=13$, $4+7+13=24$. The next term is $7+13+24=44$.

Among the numbers 35, 49, 50, 63, and 140, the number 50 is different because it's the only even number. All the other numbers (35, 49, 63, 140) are odd numbers.

The number 372164 is different because it's divisible by 11 (372164 ÷ 11 = 33833), while the other numbers are not divisible by 11. This is a divisibility test question.

The number 3781 is different because it's the only number where the product of the first two digits equals the sum of the last two digits (3×7=21, 8+1=9, not equal). The pattern for others is: 7×3=21 matches 5+9=14 (sum of digits of 21 is 2, but wait - the actual pattern is that for 7359, 1593, 9317: first digit × second digit = third + fourth digit's digits sum... actually, the simpler pattern is 3781 is the only one not following the digit rotation pattern present in others).

The number 23 is different because it's the only non-prime number in the list (23 = 3 + 2 + 3 = 8, but that's not the test - actually 23 IS prime). Wait, let me reconsider: 23 is divisible by 23 itself (which all are). The actual pattern is that 23 is the only number that is NOT divisible by 3 when summing its digits (2+3=5, not divisible by 3), whereas 13 (1+3=4), 17 (1+7=8), 37 (3+7=10), 73 (7+3=10) - this doesn't work either. The correct pattern: 23 is the only number whose digits don't multiply to give an odd number... no. Actually, the simplest pattern is 23 is the only number where the product of digits is not odd (2×3=6, even), while for others: 1×3=3 (odd), 1×7=7 (odd), 3×7=21 (odd), 7×3=21 (odd).

27 ($3^3$), 64 ($4^3$), and 125 ($5^3$) are perfect cubes. 144 ($12^2$) is a perfect square but not a perfect cube, making it the odd one out.

21, 69, and 81 are composite numbers. 83 is a prime number, making it the odd one out.

The number 121 is different because it's the only perfect square in the list (11² = 121). The other numbers (137, 153, 177, 183) are not perfect squares.

17, 29, 37, and 59 are all prime numbers. 27 is a composite number ($3 \times 9$), making it the odd one out.

For the first option, if $x=2$, then $3x(7x+12) = 3(2)(7(2)+12) = 6(14+12) = 6(26) = 156$, not 94. Thus, this equation is incorrect, making it the correct choice.

81 ($9^2$), 16 ($4^2$), and 144 ($12^2$) are perfect squares. 27 is a perfect cube ($3^3$) but not a perfect square, making it the odd one out.

The years 2000, 2012, and 2008 are leap years (divisible by 4). The year 2010 is a non-leap year, making it the odd one out.

The formula for the area of a circle is $\pi r^2$. The expression $2\pi r$ represents the circumference of a circle, making this statement incorrect.