(i) Complete the table.
| Figure number | Number of white hexagons | Number of grey hexagons | Total number of hexagons |
|---|---|---|---|
| 1 | 1 | 0 | 1 |
| 2 | 1 | 1 | 2 |
| 3 | 2 | 1 | 3 |
| 4 | 2 | 2 | 4 |
| ... | ... | ... | ... |
| 12 | 6 | 6 | 12 |
(ii) Find an expression, in terms of n, for the number of grey hexagons in Figure n.
(iii) Which figure has the same number of white hexagons as Figure 156?
Answer (2)
To help with your question, let's address each part step by step.
(i) Complete the table:
Based on the given patterns from the table, we can observe the following:
The number of white hexagons alternates between two numbers: 1 and 2.
The number of grey hexagons also alternates between two numbers: 0 and 1.
The total number of hexagons in each figure is equal to the figure number.
The pattern provides us:
For odd figure numbers: 1 white hexagon and figure number minus 1 grey hexagons.
For even figure numbers: 2 white hexagons and figure number minus 2 grey hexagons.
By following this logic, the 12th figure would have:
6 white hexagons (since 12 is even),
6 grey hexagons (total 12 - 6 = 6).
(ii) Find an expression, in terms of n, for the number of grey hexagons in Figure n:
To find a general expression, consider these patterns:
For even n: the number of grey hexagons is ( n − 2 ) /2 .
For odd n: the number of grey hexagons is ( n − 1 ) /2 .
Thus, the expression for the number of grey hexagons in Figure n (for any n) can be determined as: Number of grey hexagons = ⌊ 2 n − 1 ⌋ where ⌊ x ⌋ denotes the floor function, which returns the greatest integer less than or equal to x.
(iii) Which figure has the same number of white hexagons as Figure 156?
Following the observed pattern, for even number figures like 156, there are always 2 white hexagons.
To find another figure with 2 white hexagons, we can see all other even numbers share this characteristic.
Therefore, any even figure number, such as 2, 4, 6, 8, etc., will also have 2 white hexagons.
The simplest example would be Figure 2, which has the same number of white hexagons as Figure 156.
The table can be completed by identifying the patterns in the hexagons. An expression for the number of grey hexagons can be derived as either 2 n for even or 2 n − 1 for odd values of n. Any even-numbered figure, such as Figure 2, will have the same number of white hexagons as Figure 156, which is 2.
;
