The Necklace Code

At a glance

• Part 1: Position 19 contains a sapphire.
• Part 2: The necklace contains 5 rubies, 10 sapphires, and 4 emeralds.

Key idea

The four-jewel pattern is ruby, sapphire, sapphire, emerald. Instead of drawing all 19 jewels, separate 19 into complete groups of 4 and the jewels left over.

19 =4 ×4 +3

This means the necklace contains 4 complete pattern groups followed by the first 3 jewels of the next group.

1. Identify position 19

The 4 complete groups fill positions 1 through 16. Positions 17, 18, and 19 restart the pattern:

1. Position 17 is ruby.
2. Position 18 is sapphire.
3. Position 19 is sapphire.

Therefore position 19 contains a sapphire.

2. Count each jewel type

One complete group contains 1 ruby, 2 sapphires, and 1 emerald. Four complete groups contain:

• 4 ×1 =4 rubies;
• 4 ×2 =8 sapphires; and
• 4 ×1 =4 emeralds.

The final three jewels are ruby, sapphire, sapphire. Adding those gives 5 rubies, 10 sapphires, and 4 emeralds.

Check

The three counts must total the 19 positions: 5 +10 +4 =19. Position 19 is also the third position in its four-jewel group, and the third jewel of the pattern is sapphire.