Serena colours the hexagons on the tiling shown. If two hexagons share a side, she colours them with different colours. What is the least number of colours that she can use to colour all of the hexagons?
(A) 4
(B) 6
(C) 7
(D) 2
(E) 3

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-044.jpg?height=288&width=325&top_left_y=2187&top_left_x=1323)
Since we seek the minimum number of colours that Serena can use to colour the hexagons, we first determine if it is possible for her to use only two colours (using only one colour is not possible). We will use the numbers $1,2,3$ to represent distinct (different) colours.

We begin by choosing any group of three hexagons in which each pair of hexagons share a side, as shown.

We colour two of the hexagons shown with colours 1 and 2 (since they share a side).

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-215.jpg?height=154&width=146&top_left_y=2189&top_left_x=1575)

Each of these two coloured hexagons share a side with the third hexagon which therefore can not be coloured 1 or 2 .

Thus, the minimum number of colours that Serena can use is at least three.

Next, we determine if the entire tiling can be coloured using only three colours.

One possible colouring of the tiles that uses only three colours is shown to the right.

While other colourings of the tiles are possible, Serena is able to use only three colours and ensure that no two hexagons that share a side are the same colour.

There are many nice patterns of the colours in this tiling.

Can you find a different colouring of the tiles that uses only three colours?

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-216.jpg?height=363&width=385&top_left_y=236&top_left_x=1485)

ANSWER: (E)

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