Wash the beets and cut into quarters and place in a saucepan with 6 cups of cold water and the vinegar. Bring to a boil, reduce heat and simmer for 20 minutes. Remove from heat.
While the beets are cooking, boil the eggs; place the eggs in a pot in a single layer. Cover with cold water by one inch. Bring to a boil and as soon as the water comes to a boil, remove from the heat and cover the pan. Let the eggs sit in the water for 12 minutes. Remove the eggs from the hot water, crack the shells well but leave the shells on the eggs. Place the warm eggs in the warm beet liquid and let the eggs sit for 30 minutes. Remove the eggs and discard the beets and beet liquid.
Peel the eggs under cold running water. Dry and cut in half lengthwise. Remove the yolks, place in a mixing bowl and mash with a fork. Add the mayonnaise and wasabi and mix well. Pipe or spoon the egg yolk mixture back into the whites. Garnish with olives if desired. Refrigerate until serving.