A
triomino (not to be confused with a
tromino) is an equilateral triangle-shaped tile with numbers in each corner. Like the game of dominoes, the object is to place them next to each other so their numbers match.
The standard set I own with numbers 0 to 5 has 56 tiles. Does the company care about orientation?
Find two formulas for the number of triominoes in a set numbered from 0 to n both considering and not considering orientation.
For a set with numbers from 0 to n:
Triominoes with only one number repeated:
There are n+1 possible of such triominoes.
With two different numbers:
There are n+1 choices for the unique number and then n choices for the duplicated number, for n*(n+1) possible triominoes.
With three different numbers:
There are C(n+1,3) possible selections of the three numbers. If orientation (order) does not count, this is also the number of possible such triominoes. If orientation counts, that number is doubled.
In the case of n=5:
n+1 6
n*(n+1) 30
C(n+1,3) 20
----
56
The company does not care about orientation; if it did, there would be another 20 triominoes--the reversals of all 20 that have three different numbers on them.
The formulas:
order doesn't matter: (n+1)^2 + C(n+1,3)
order does matter: (n+1)^2 + 2*C(n+1,3)
|
|
Posted by Charlie
on 2021-10-13 11:42:39 |
solution
