Place the beets into a saucepan, cover with water, and bring to a boil. Reduce heat to medium, and boil gently until tender, about 20 minutes. Remove from the boiling water and let cool. Peel and dice the beets, and chill.
Place the potatoes into a separate saucepan, add water to cover, and stir in salt. Boil the potato cubes until tender but not mushy, about 15 minutes. Drain the potatoes in a colander set in the sink, and chill.
Bring a saucepan of water to a boil, reduce heat to medium, and stir in 1 to 2 more teaspoons of salt. Very gently lower the eggs into the boiling water, making sure they don’t hit the bottom of the pan and crack; simmer for 15 minutes. Remove the eggs to a bowl filled with ice and ice water, and let the eggs chill until thoroughly cold. Peel and dice the eggs.
Mix the chilled beets, potato cubes, and eggs together in a salad bowl, and gently stir in mayonnaise to taste.
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