Consider the following figure :
[Image of a circle divided into 6 sectors with numbers 1, 2, ?, 8, 4, 12 in sectors clockwise, starting from top right]
Find out the missing number from among the following :
Let’s look for patterns between adjacent numbers or opposite numbers.
Examining opposite numbers:
1 is opposite 8. Product = 1 * 8 = 8.
2 is opposite 4. Product = 2 * 4 = 8.
The missing number (?) is opposite 12. If the pattern is that the product of opposite numbers is constant (8), then ? * 12 = 8, which means ? = 8/12 = 2/3. However, 2/3 is not among the integer options.
Let’s examine the relationship between adjacent numbers. Let P(i) be the number at position i (clockwise, starting from top-right as position 1).
P1=1, P2=2, P3=?, P4=8, P5=4, P6=12.
P2 = P1 * 2 (1 * 2 = 2)
P5 = P4 / 2 (8 / 2 = 4)
P6 = P5 * 3 (4 * 3 = 12)
P1 = P6 / 12 (12 / 12 = 1)
We have the operations: x2, ?, ?, /2, x3, /12.
Let the unknown operations be xM1 and xM2:
P2 = P1 * 2
P3 = P2 * M1
P4 = P3 * M2
P5 = P4 * (1/2)
P6 = P5 * 3
P1 = P6 * (1/12)
Let’s try the options for ?.
If ? = 16 (Option B), then P3 = 16.
P3 = P2 * M1 => 16 = 2 * M1 => M1 = 8.
P4 = P3 * M2 => 8 = 16 * M2 => M2 = 8/16 = 1/2.
The sequence of multipliers between adjacent numbers becomes: 2, 8, 1/2, 1/2, 3, 1/12.
1 x 2 = 2
2 x 8 = 16
16 x 1/2 = 8
8 x 1/2 = 4
4 x 3 = 12
12 x 1/12 = 1
This sequence of operations links all numbers in the circle correctly when the missing number is 16. While the sequence of multipliers (2, 8, 1/2, 1/2, 3, 1/12) isn’t trivially patterned, it allows all given numbers and one option to fit consistently. The alternative pattern of opposite products (8) gives a non-integer result not among options. Therefore, 16 is the most likely intended answer based on finding a consistent (though complex) relationship between adjacent numbers that incorporates one of the options.
– The pattern might involve multiplication or division.
– Test options if a simple pattern isn’t immediately obvious.