Why small states are dots
The quiz map uses Natural Earth data at a scale that suits a whole-world view. At that scale 29 countries are smaller than a single point, so they are drawn as round markers instead of outlines. A few more countries have a shape but get a dot as well whenever they are smaller than about six pixels on your screen; zoom in and the dot gives way to the real outline.
How to tap them
A tap right on a dot always takes the dot, even when a larger country lies underneath. If you tap next to it, the country under your finger counts instead. On a phone it is worth zooming in two or three steps before tapping in the Caribbean, the Persian Gulf or the Pacific, where several dots sit close together.
Use the hint well
In the map mode the hint draws a ring around a wider area that contains the country and zooms there. For small states that is often all you need: the dot inside the ring is the only candidate. The hint halves the points, which is a fair price for a country the size of a town.
Practise them as a set
States under 3,000 km² have a set of their own in every mode. Try them in find the country first, then in name them all: typing all 29 against the clock is a good test of whether the map has stuck.