Math Worksheets

Question: Penny Triangle Equation?

Alright so for my math class we got a worksheet where we have a Triangle of a certain amount of pennies that are in the shape of a Triangle and then we have to find out how many moves it takes to turn the Triangle upside down.

it says that we need to write an equation for the least number of pennies you would need to move in order to flip a penny Triangle made from N pennies on a side. the rule is not N-1.

i figured out that on a Triangle made of 3 pennies it takes 1 move.

on a Triangle made of 6 pennies it took 2 moves

and on a Triangle made of 10 pennies it took 3 moves.

can someone please help me find the equation?

Answer: The sequence continues:
Length of side of Triangle:
1, 2, 3, 4, 5, 6, 7, .8, 9, 10, 11, 12, 13, 14, 15
Number of moves required
0, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40

It goes up by 1’s (1,2,3)
then by 2’s (5, 7, 9)
then by 3’s (12, 15, 18)
then 4’s, 5’s, 6’s and so on

What happens is:
First you can move 1 penny from (1, 2, 3) corners of the Triangle.
Then you have to move
1 penny from two corners and 3 from the the other one,
then one 1 and two 3’s,
then three 3’s.

Now you’re out of 3’s, so you go on to 6’s
two 3’s and one 6
one 3 and two 6’s
zero 3’s and three 6’s,

then 6’s and 10’s
10’s and 15’s
and so on.

Here’s the full scoop:
Sequence A001840 in the O E I S

http://www.research.att.com/~njas/sequences/A001840

The simplest of the formulas (from Formula section there) is in three parts:
a(3k-1) = k (3k+1) / 2
a(3k) = 3k (k+1) / 2
a(3k+1) = (k+1) (3k+2) / 2

Or you can have a recursive formula:
a(0) = 0, a(1) = 1, a(2) = 2
a(n) = a(n-3) + n

It’s not simple, as you can see!

Comparing and Ordering Integers – YourTeacher.com – Math Help