A rep-tile is a shape that can tile larger copies of the same shape.

**Question 1:** Are there any convex pentagons that are also rep-tiles?

**Remarks:** 15 convex pentagonal tiles of the plane are known and none of them *appears* to be a rep-tile. Assuming this observation is right, one can invoke a proof given in 2017 by Michel Rao - that these 15 are the *only* convex pentagonal tiles possible - to answer our question in the negative. However, I don't know if Rao's proof has been validated and if there is a simpler (elementary) proof that there are say no convex pentagonal rep-tiles. So, here one is actually asking if there is a simpler proof for a weaker claim.

**Definition:** Let us say a *multi-way rep-tile* is a polygon *P* with the property: if *P1* and *P2* are magnified copies of *P* and *P1* can be tiled with *m* units of *P* and *P2* can be tiled with *n* units of *P* with *n*> *m*, then, a layout of *n* units can form *P2* *without* *m* of the units in the layout together forming a *P1*. As shown on this page: https://en.wikipedia.org/wiki/Rep-tile, there are isosceles trapeziums with multi-way property (with *m*= 4 and *n* = 9). On the other hand, the square is obviously a rep-tile but *not* multi-way.

**Question 2:** Are there other convex polygons with this multi-way rep-tile property?

wasn'ton the list of the 15 families that tile the plane periodically turned out to be a rep-tile, then that induced tiling would per force have to be aperiodic. $\endgroup$doesconsider aperiodic tiles, and finds none - see e.g. this Quanta article for a description of the classification strategy, which proceeds by trying local patches until a contradiction is reached rather than assuming periodicity. $\endgroup$2more comments