diff --git a/academic/misc/2024-math-calendar/12-dec.html b/academic/misc/2024-math-calendar/12-dec.html index 732083e..0589aa5 100644 --- a/academic/misc/2024-math-calendar/12-dec.html +++ b/academic/misc/2024-math-calendar/12-dec.html @@ -733,6 +733,159 @@

Dec 15

Dec 16

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + How many ways can + you tile the shape with + 2 × 1 dominoes? + + + +

+First consider a domino along the thin edge. There are 2 possibilities shown +below. +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+In the first possibility, placing a domino adjacent on the left of the red tile +leaves a single space that cannot be filled with a \(2×1\) domino. So we +use the second possibility. By applying similar reasoning to the other 4 edges, +we must have dominoes in the following positions. +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+This leaves 4 \(2×2\) spaces. Each of these has 2 possibilities: 2 +vertical dominoes or 2 horizontal dominoes. The total number of possible tilings +is \(2^4=16\). +

+

Dec 17

Dec 18