Question

Hi, I would appreciate any help for this problem I don't really understand for discrete math. Thanks! (:

15. (P5) Remember, an L-tromino is a shape consisting of three equal squares joined at the edges to form a shape resembling the capital letter L. Consider the following "theorem" "Theorem": For any integer n 1, if one square is removed from a 2·2" × 3·2" checkerboard. the remaining squares can be completely covered by L-shaped trominoes. What follows is a supposed proof. Your job is to critique the proof by explaining what is wrong with it: "Proof": We prove the theorem by mathematical induction Assume it is true for k that a 2-2'x3-2' checkerboard with one square missing can be completely covered by L-shaped trominoes. We will show this implies it is true for k +1 that a 2-2k+1 x3-2k+1 checkerboard with one square missing can be completely covered by L-shaped trominoes. Note that a 2-2k+1 x3-2k+1 can be split into four quadrants, each one a 2-2*x3-2k checkerboard The missing square on the 2-2k+lx3-2k+1 checkerboard must occur in one of these four quadrants By the induction assumption, this quadrant, which is of size 2-2*x3-2* and missing one square, can be completely covered by L-shaped trominoes. Now, place one tromino at the intersection point so that it covers one square from each of the These three quadrants are each of size 2-2kx3-2k and missing one square. Thus, by the inductiorn Therefore, we have shown that the entire 2-2k+1x3-2*+1 checkerboard with one missing square Hence, we have proved our theorem by mathematical induction. QED, bro! other three quadrants, assumption, they can be completely covered with L-shaped trominoes. can be completely covered by L-shaped trominoes.

Answer 1

e proveb Prine pafatb emoth col Induetion ale prove the statement for n-1厂Th,sismissing uchon .S n=lr the checl-board has size For n - e the check board as total 24 oquare s If we remove one seware rom hen 23 squares are remining A 1- shaped tro m contains 3- squares a ghope is telyAll ed by L-shaped trominoes comple no.o Squares Squarein one Irominors oving one-square弁om 212 X3x2, checkhatrcl e d trominot eanno cem plete'y as it not correr4na). nd the proct given proof cannot be Correc」允r n- by malhema tica inductoni not correct aas not proved