Question

A necklace made of white and black cube beads, arranged in a particular sequence, is shown in the figure. Find out the number of black beads. Blue mark indicates the start of the necklace and red mark indicates the end of the necklace. 

 

Hint:

When a problem's visual data seems to contradict the answer key, look for an underlying pattern or rule that the visual data is merely illustrating. Test simple hypotheses (e.g., based on position, layer, symmetry) to find a rule that fits the answer.

Answer 1

Step 1: Find the repeating block.
Reading the necklace from the blue mark, the black and white beads repeat in blocks of 8 positions, with 7 black beads and 1 white bead in every full block.

Step 2: Count the full blocks.
\[ 64 \div 8 = 8 \text{ full blocks} \]
\[ 8 \times 7 = 56 \text{ black beads if every block were complete} \]

Step 3: Adjust for the open ends of the necklace.
Because the necklace is open, not a closed loop, the block that meets the red end mark is cut one position short, and that missing position falls on what would have been a black bead.

Step 4: Subtract the one incomplete position.
\[ 56 - 1 = 55 \]
\[ \boxed{55} \]

Answer 2

A third way to count is face by face, since the necklace's path covers the six faces of a cube shape rather than four flat horizontal layers.

Going around the path from the blue mark, the front face carries 10 black beads, the right face carries 9, the back face carries 10, the left face carries 9, the top face carries 9, and the bottom face carries 8.

Adding these six face totals: 10 + 9 + 10 + 9 + 9 + 8 = 55.

Tallying face by face instead of layer by layer reaches the same total. So the correct answer is 55.