I was given “The Big Book of Brain Games” by Ivan Moscovich for Xmas. Most are too easy but here is a nice one (number 331):
Construct a square from four identical linkages hinged at the corners. Such a figure is capable of moving on its hinges to become a rhombus. How many linkages of the same length must be added to make the square rigid? The linkages must be in the same plane as the square and each one can be connected only at the hinges.
My best solution so far has 43 extra linkages which must be far too many.
Update 28-Dec-2010: Lubos has given a nice solution with no overlapping links which requires only 31 extra edges or 29 if you allow the links to cross. However I have found out that this is still not the best solution for the case where overlaps are allowed! so keep trying.
Final Update: Since posting this puzzle I have learnt that a version of it was posed in Martin Gardner’s SciAm column in 1963. His version required that the bracing links do not overlap. Seven readers sent in the solution with 23 added links shown below.
Erich Friedman considered the case where links can cross in 2000 and posted results on his Math Magic website. His best solution had 17 extra links. However, someone later informed him that Andrei Khodulyov had found a solution some time ago with just 15 extra links.
Well done to all those who posted solutions here and over at The Reference Frame.