$9$ hours.

$$ \frac{\text{time taken to dig a bed of diameter six metres}}{\text{time taken to dig a bed of diameter four metres}}=\frac{6^2}{4^2}=\frac{3^2}{2^2} = \frac{9}{4}$$

$$ \frac{\text{time taken to dig a bed of diameter six metres}}{\text{time taken to dig a bed of diameter four metres}}=\frac{6^2}{4^2}=\frac{3^2}{2^2} = \frac{9}{4}$$

Hence the time taken for one gardener to dig a bed of diameter
six metres is $ \frac{9}{4} \times4 \; \text{hours} = 9
\; \text{hours}$

The time taken for six gardeners to dig six flower beds is the same as the time it takes one gardener to dig one bed of the same size, and hence the answer is $9$ hours.

*This problem is taken from the UKMT Mathematical Challenges.**View the archive of all weekly problems grouped by curriculum topic*