A necklace is to be created that contains only square, circular, and triangular beads. A total of \(180\) beads will be strung on the necklace as follows. First, \(1\) square, then \(1\) circular, and then \(1\) triangular bead are strung on the necklace. Then, \(2\) square, then \(2\) circular, and then \(2\) triangular beads are strung on the necklace. This pattern continues, with each time there being one more of each type of bead strung on the necklace. So, next there are \(3\) square, then \(3\) circular, and then \(3\) triangular beads strung on the necklace, then \(4\) square, then \(4\) circular, and then \(4\) triangular beads, and so on.
Once the necklace is completed, how many of each bead shape will be on the necklace?
An equal number of square, circular and triangular beads occur after \(3(1) = 3\) beads are strung, then after \(3(1) + 3(2) = 9\) beads are strung, then after \(3(1)+3(2)+3(3)=3+6+9=18\) beads are strung, and so on.
The greatest total number of beads that can be strung while maintaining equal numbers of squares, circles and triangles is: \[\begin{aligned} 3(1)&+3(2)+3(3)+3(4) + 3(5) + 3(6) + 3(7) + 3(8) +3(9)+3(10)\\ &=3+6+9+12 + 15 + 18 + 21 + 24+27+30\\ &=165 \end{aligned}\]
At this point, there are \(165\div 3=55\) of each of the three shapes. The final group of \(30\) beads strung are \(10\) squares, \(10\) circles and \(10\) triangles, in that order.
The next group would contain \(11\) of each shape if all of the beads could be strung. However, there are only \(180-165=15\) beads remaining to be strung. The first \(11\) beads in this group are squares, leaving \(4\) beads to be strung. At this point there are \[55+11=66\] square beads on the necklace. The final \(4\) beads strung are circles, giving a total of \[55+4=59\] circular beads on the necklace. No more triangular beads can be added, so the total number of triangular beads remains at \(55\).
Therefore, the completed necklace will contain \(66\) square beads, \(59\) circular beads, and \(55\) triangular beads.