Mathematical Chunks of Sentient Protoplasm (MCSPs, for short) are smart blobs who dream of merging together into one huge blob. But they can only do it following certain rules: If two MCSPs have the same mass, or if their masses are 1 apart, they can merge into a single MCSP, whose mass will be the sum of the original two. If an MCSP has even mass, it can split into two MCSPs, each with half the original mass. Suppose we start with n MCSPs, with masses 1 through n. For what values of n is there a finite sequence of steps that will allow all n MCSPs to merge together into a single MCSP and achieve their dream of unity?

An electronics manufacturer wants to know if customers would be interested in a detachable keyboard for their tables and if so, what the most important features would be. the manufacturer partners with an electronics store to include copies of the survey with every purchase at that store for one week. which of the following best explains why this random sample is unlikely to be a good representative sample of tablet owners' preferences for a detachable keyboard? a) one week is likely not enough time to get a large enough sample. b) most people won't bother to send in the survey, which is likely to skew the results c) there is no way to verify whether the responders to the survey actually own a tablet d) the survey is biased because it was conducted through an electronics store, not the general population

Polygons efgh and e′f′g′h′ are shown on the coordinate grid: what set of transformations is performed on efgh to form e′f′g′h′? a. a translation 1 unit to the left followed by a 90-degree counterclockwise rotation about the origin b. a translation 1 unit to the right followed by a 90-degree counterclockwise rotation about the origin c. a 90-degree clockwise rotation about the origin followed by a translation 2 units to the right d. a 90-degree clockwise rotation about the origin followed by a translation 2 units to the left

Identify reflection rule on a coordinate plane that verifies that triangle a(-1,7), b(6,5), c(-2,2) and a’(-1,-7), b(6,-5), c’(-2,2) triangle are congruent when reflected over the x-axis