The plants flower and then the flowers die and the plants start to go yellow and die off, then the potatoes are ready

Joy is right to a degree....it depends on the variety....some don't have flowers. A useful rule of thumb is that they should be ready 12 weeks after planting....the best way to find out is to dig one up......or if they are in pots/bags...put your hand down the side of the bag/pot and see if the spuds are large enough......

Joy is correct if a little simplistic. Early potatoes should be ready around eight weeks after planting, so for us it is July time. 2nd earlies take a bit longer whilst main crop potatoes for us at least are late August early September. Not all tatties flower! As a test scrape away the soil from the side of a ridge of first early potatoes and see if you can see any reasonable sized tatties. Remember that first early tatties are not going to be as big as main crop tatties. Also, first early potatoes will only keep for a short time so harvest as needed.