Heat water (salted after starts boiling) to boiling in 3-quart saucepan. Add potatoes. Cover and heat to boiling; reduce heat. Cook 20 to 25 minutes or until just tender; drain. Cool slightly; peel and cut into cubes. Mix mayonnaise with celery, pepper, radishes and cucumber. While potatoes are still hot add to vegetables and gently toss. Stir in eggs. Cover and refrigerate at least 4 hours until chilled.