The problem asks us to count the number of beads hidden inside the box, given a string of beads that follows a certain pattern.

1. Problem Description

The problem asks us to count the number of beads hidden inside the box, given a string of beads that follows a certain pattern.

2. Solution Steps

First, let's identify the pattern of the beads. We can see a pattern of 3 blue beads followed by 1 white bead.

Then we count the number of blue and white beads that are visible outside the box. On the left side of the box, we have 3 white beads and 8 blue beads. On the right side of the box, we have 1 white bead and 7 blue beads.

Now we count the total number of beads visible. $3 + 8 + 1 + 7 = 19$ visible beads.

If the pattern is followed, the number of beads from the left side to the right side are alternatingly 3 blue beads, 1 white bead. The total number of beads is $4n$ if we have $n$ repetitions of the pattern. Let's continue the pattern.

On the left side, starting from white we have "white, blue, blue, blue, white, blue, blue, blue, white, blue, blue" (3 white, 8 blue). The next color would be blue if the beads were visible, so the hidden beads would be 3 blue beads followed by 1 white bead, and that pattern will continue in the box. We keep counting till it reaches to the end of box where we have 7 blue beads and 1 white bead visible.

Let $x$ be the number of blue beads and $y$ be the number of white beads hidden inside the box. Then, the entire string has $8+x+7$ blue beads, and $3+y+1$ white beads. Since 3 blue beads are followed by 1 white bead, the entire number of beads inside the box should follow this pattern.

The beads outside are: 3 white, 8 blue, then some beads in the box, then 7 blue and 1 white. Total: 4 white + 15 blue + unknown beads. Consider repeating pattern blue blue blue white. In the box, we should have blue blue blue white blue... .

Let us count the number of beads just before the box starts. It goes white, so we expect blue, blue, blue inside the box first. On the other end, we have blue, blue, ...blue white, so the next one coming outside box is a "blue". So, within the box, there are "white, blue, blue, blue" for number of rounds. The arrangement inside the box should have 1 white and 3 blue or 2 white and 6 blue or 3 white and 9 blue or n white and 3n blue in general. The number of beads in total in the box is 4n.

On the left side, there are 3 white and 8 blue. Next will be "blue".

On the right side, there are 1 white and 7 blue. The last is white, so we add 3 blue.

Now we sum up, 3 white, 8 blue, then we need blue, blue, blue, white such that at the other end we have blue.

The beads in the box are: blue blue blue white blue blue blue white blue = 9blue and 2 white, total

1

1. Let's try

1

8. We have 3 white + 18 + 1 white = 4 white. We have 8 blue +18+7 = 15+18=33 blue. 4:

3

3. This does not match pattern

We need to figure how many beads are inside the box.

Let's consider if 16 is inside the box. Then, 3 white+1 white = 4 white. Blue 8+7 = 15 blue. 4+15+16 =

3

5. We know that the pattern is 3 blue and 1 white.

Suppose 18 beads inside the box. We have 15+18 = 33 blue and 4 white. 33+4 =

3

7.

3+1 + beads in box = some number

If total count is $4k$,

Visible Beads are 3 white and 8 blue.

We see 19 visible, thus assume 1 bead in the box means sum = 20 which = 4k, then k =5 so must be correct.

3. Final Answer

B) 17