Tilings math challenge

[enthumath]

Isabel was making a rectangular strips 2 units wide using 2 x1 tiles. She was interested in the number of different strips of each length she could make. Here are some of her examples arranged according to units of length. Isabel regards all the strips shown as different even though some are flips of others.



a) draw the eight ways of making strips 5 units long- (i have already solved this)

Isabel noticed that the number of ways of making strips followed a pattern: the number of strips n units long is always the sum of the numbers of the strips of the two previuos lengths. By way of example, if Wn is the number of strips of length n the diagrams above show that W4= W3 + W2 since 5= 3+2. In general her guess is that

W1= 1, W2= 2 and W1= W(N-1) + W(N-2) when n>or= 3

b) explain why Isabel's huess is true
c) Isabel experiemnted with 3 x 1 tiles, making strips 3 units wide. Find a formula similar to that above, for the number of strips of length n

d) Find a similar formula for the number of ways of making strips of width m and length n from mx1 tiles

I would appreciate your help- Thank you very much

[am99]

A is stupidly easy but i am having a hard time on b and c and d but i have found the pattern a, b, a+b, (a+b)+b, ((a+b)+b) + (a+b).
example 1, 2, 3, 5, 8, 13 i have proven this but idk wat to do now

[unknown2006]

Can someone help me with b, c, d and explain it please soon

[alkatran]

intuitively:

b: is true because you can only start the sequence two ways: a vertical bar or two horizontal bars.

c: similar to b except this time the two ways are a vertical bar and three horizontal bars, so the formula falls right out that

d: obvious almost right away after b and c are done

[M_G_101]

Just a word...well hints you could say...
With 5 b i will tell you something that may end up helping you with understanding the equation 'Isabel' has...
W stands for the number of tiling patterns you can make and the little n is how wide they have to be i.e. W3 is the number of ways you can make tiling patterns that are 3 units wide so W1=1, W2=2, W3=3, W4=5, W5=?
-- Use the formula for W5--
Oh... and MAYBE Fibonacci has something to do with it...
Hope this helps! No doubt it helped me...
P.S. This is exactly the type of hint that a teacher gave me so i am not repeating anything that can't be said...

[unknown2006]

still having trouble 5 c, d

[M_G_101]

Well with 5c... it REALLY helps if you draw the tiling combinations at least up to 8 units wide... then you should get the equation for that...
5d requires you to draw more diagrams... they REALLY help!! No joke... i don't think you can do it without em... I did 4 x 1 strips and 5 x 1 strips and managed to get the answer and it is really clever when you figure it out... and don't worry... it may seem repetitive in the beginning but if you continue the sequence for long enough you are SURE to get the answer Hope this helps!!!

[unknown2006]

still don't get it