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$14 \; \text{cm}$. The length of $AD$ must be less than $15 \; \text{cm}$, since $15 \; \text{cm}$ would be its length if all four points lay in a straight line. However, by making angles $ABC$ and $BCD$ close to $180^{\circ}$, $AD$ can be made close to $15 \; \text{cm}$ in length.

As the length of $AD$ is a whole number of centimetres, its maximum value, therefore is $14\; \text{cm}$

This problem is taken from the UKMT Mathematical Challenges.