is it possible to arrange 5 coins so that every coin touches every other coin?
coins have to lay flat on the table, and coins cannot be stacked
It depends on what you mean by "flat" and stacked.
http://www.nada.kth.se/kurser/kth/2D1431/02/lecture11_sl.pdf
To determine if it is possible to arrange 5 coins so that every coin touches every other coin, you need to consider the geometry of the situation. One approach is to draw a diagram or use a physical representation to visualize the possible arrangements.
A helpful resource for understanding such arrangements is a math concept called graph theory, specifically the concept of a complete graph. In a complete graph, every vertex (representing a coin) is connected to every other vertex (coin) by an edge. In this case, the edges between the coins would represent the touching points.
Using graph theory, we can determine that a complete graph with 5 vertices requires 10 edges. Each coin can connect to 4 other coins, but since the coins cannot be stacked or overlap, they can only touch each other at a single point.
Drawing the possible arrangements, you will find that it is not possible to arrange 5 coins so that every coin touches every other coin if they lay flat on the table and no stacking is allowed. This is because a complete graph with 5 vertices would require 10 edges, but only 5 points of contact are available when the coins are laid flat.
To visualize and understand this concept better, you can refer to the provided link to a lecture on complete graphs in graph theory.