A busy hive is growing one ring at a time. Four stages are already built. Spot the pattern in the cell counts and work out the next one before you peek.
The puzzle
Each hive starts with the one before it and adds a complete new ring of hexagon cells all the way around the outside. The cell counts so far are:
- Hive 1: 1 cell
- Hive 2: 7 cells
- Hive 3: 19 cells
- Hive 4: 37 cells
- Hive 5: ?
How many cells will Hive 5 have?
Show hint
Don’t count the whole hive. Count only the cells in each new ring, and see how that number changes from one hive to the next.
Show answer
Answer: 61.
- Find how many cells each new ring adds: 7 − 1 = 6, 19 − 7 = 12, and 37 − 19 = 18.
- The rings grow by 6 cells each time (6, 12, 18), because a hexagon has six sides and each side of the new ring is one cell longer than the last.
- So the new ring around Hive 5 adds 18 + 6 = 24 cells.
- Hive 5 has 37 + 24 = 61 cells.
Check with the shortcut: Hive n has 3 × n × (n − 1) + 1 cells. For n = 5, that is 3 × 5 × 4 + 1 = 61. These are called centered hexagonal numbers, and the middle row of Hive 5 is 9 cells wide.