Question

The boxes are arranged in a unique number pattern. Which numbers will be on the green, orange, and yellow gift boxes?

Answer 1

The boxes are arranged so that the numbers form what is called as a Pascal's triangle, i.e., it follows the pattern as described below.

Each number in the arrangement is either a 1 if it does not have 2 numbers directly above it, or the sum of the two numbers directly above it if it has 2 such numbers.

i.e.,

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1

and so on.

Using this pattern, we deduce that the numbers on the green, orange and yellow boxes are 1, 3 and 10 respectively.

Alternatively, we also observe that in a Pascal's triangle, the horizontal sum will double each time we proceed to the row below, i.e.,

For instance,
1 --> Sum is 1
1 1 --> sum is 1x2 = 2
1 2 1 --> Sum is 2 x 2 = 4
1 3 3 1 --> Sum is 4 x 2 = 8
and so on....


Therefore, in the second row (green box):

let the missing number be x. Then

1 + x = 2

So, x = 1

Similarly, let the number in the orange box be y. Then

1 + 3 + y + 1 = 8

So, y = 3

Now, let the missing number in the 5th row be k.

Using the rule:
"Each number is the sum of the 2 numbers directly above it, if there are 2 such numbers."
we get:
y + 1 = k
3 + 1 = k

So, k = 4.

Now consider the 6th row:
Let the missing number in the yellow box be m.
Using the rule:
"Each number is the sum of the 2 numbers directly above it, if there are 2 such numbers." again,
we get
6 + k = m
6 + 4 = m

Thus, m = 10.

Hence, the numbers on the green, orange and yellow boxes are 1, 3 and 10 respectively.

Option b will be the correct answer.