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The solution here depends on using Pythagoras theorem in 3 dimensions (actually using the theorem for a right angled triangle on the floor and then a second time for a vertical right angled triangle). The centre of the large ball (of radius $5$ cm) is at the centre of the box and, if you think of the straight line from one corner of the box to the centre of the box, it goes through the centre of a small `packing' ball.
This solution came from Christopher, Bishop's Stortford College.