Slice the unpeeled potatoes. Cook in a small amount of boiling salted water for 15 to 20 minutes or until tender. Drain and cool potatoes.
In a very large mixing bowl, combine potatoes, cucumber, celery, and the radishes. Chop 3 of the eggs and fold into the potato mixture.
In a small mixing bowl, combine the salad dressing, onion, mustard, and ranch dressing mix. Salt and black pepper to taste.
Add dressing to the potato mixture. Toss lightly to coat. Cover and chill for 6 to 24 hours. Slice remaining egg; arrange on salad. Sprinkle with chives, if you like. Makes 12 to 16 servings.