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.

Question

Pointing to a photograph, a man says, "I have no brothers or sisters, but the father of the person in the photograph is my father’s son." Who is the person in the photograph?

Answer 1

Step 1: Note the total bead count.
The necklace runs through all 64 positions of the 4-layer, 4x4-per-layer arrangement, from the blue start mark to the red end mark.

Step 2: Look for a repeating colour block.
Following the sequence from the blue mark, the colours repeat in the same short block of black and white beads for most of the necklace, so counting one repeating block and scaling up covers most of the 64 positions quickly.

Step 3: Adjust for the start and end of the loop.
Because the necklace has a fixed start and end (blue and red marks) rather than repeating forever, the last partial block does not match the earlier full blocks exactly, so a small adjustment is needed at the point where the sequence closes.

Step 4: Final Answer.
Once the repeating blocks and the adjusted closing section are added together, the count comes to \(\boxed{55}\) black beads.

Answer 2

A third way to approach this is to pair up the layers of the 4-layer arrangement and compare each pair, rather than counting each layer separately or counting the minority colour.

The 4 layers can be paired as (layer 1, layer 3) and (layer 2, layer 4), since the coloring pattern shown repeats in a similar way between these pairs.
Counting the black beads on one representative layer from each pair and doubling it (since its partner layer shares the same count of black beads) gives the black-bead total for that whole pair in one step, instead of counting all 4 layers individually.
Adding the totals from both pairs together accounts for every position in the necklace except the single junction bead where the sequence closes from the red mark back toward the blue mark, which needs a one-off check since it does not belong cleanly to either pair.
Once that junction bead is checked and folded into the running total, the full count of black beads comes to 55. \[ \text{Black beads} = 55 \]

So the correct answer is 55.

Answer 3

A third way to approach this is to pair up the layers of the 4-layer arrangement and compare each pair, rather than counting each layer separately or counting the minority colour.

The 4 layers can be paired as (layer 1, layer 3) and (layer 2, layer 4), since the coloring pattern shown repeats in a similar way between these pairs.
Counting the black beads on one representative layer from each pair and doubling it (since its partner layer shares the same count of black beads) gives the black-bead total for that whole pair in one step, instead of counting all 4 layers individually.
Adding the totals from both pairs together accounts for every position in the necklace except the single junction bead where the sequence closes from the red mark back toward the blue mark, which needs a one-off check since it does not belong cleanly to either pair.
Once that junction bead is checked and folded into the running total, the full count of black beads comes to 55. \[ \text{Black beads} = 55 \]

So the correct answer is 55.

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.

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Correct Answer: 55

Approach Solution - 1

Approach Solution -2

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