In discovering the answer to the problem , if Sum of square of n natural numbers formula crossed your mind, you pat yourself on your back. S = product of n x (n+1) x (2n+1) divided by 6 But what is value of n ? Let me wait for few more responses..
The basal packing contains 81 pieces of 24+20+16+12+8+1=81 spheres. Then it gets narrowed to 57,37,21,9,and 1=125 So over all spheres may be 206 . No formula used. Vathsala Jayaraman
The arrangement of spheres is in a formation called a square pyramid. Each layer of the pyramid is a square made up of spheres. There are 7 layers so n = 7 The top layer has 1 sphere, Layer 2 has 2^2 (2 squared) spheres = 4 Layer 3 has 3^2 = 9 Layer 4 has 4^2 = 16 Layer 5 has 5^2 = 25 Layer 6 has 6^2 = 36 Layer 7 has 7^2 = 49 1+4+9+16+25+36+49 = 140 As you can see the answer is the sum of the squares of the first n natural numbers where n is the number of layers in the pyramid. These kind of numbers are called square pyramidal numbers and can be computed directly using the formula you've given in your reply!