The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Stair Patterns

Date: 02/27/2001 at 07:33:23
From: pierre aghajani
Subject: Need a formula, please

Could you help me with stair patterns made with matches?

The first step is made with four matches; the second step is made from 
10 matches; the third step is made from 18 matches; the fourth step is 
made from 28 matches.

How many matches would be needed to build 6, 10, and 50 steps? 

We must also find the number of steps in the largest stair pattern 
that can be made if we have 5000 matches.  


Date: 02/27/2001 at 16:17:29
From: Doctor Jerry
Subject: Re: Need a formula, please

Hi  Pierre,

If you think about building these stairs each on the right side of the 
previous stair, then each of the bottom and top blocks in the new 
stack requires three matches, and the ones in between need only two.  
The number of in-between ones goes up by 1 each time

S1 = 4
S2 = S1+2*3     no in-between ones
S3 = S2+2*3+1*2
S4 = S3+2*3+2*2
S5 = S4+2*3+3*2
and so on.


S1 = 4
S2 = 4+2*3
S3 = 4+4*3+1*2
S4 = 4+6*3+(1+2)*2
S5 = 4+8*3+(1+2+3)*2
and so on.

In S5, the 8 is 2(5-1)*3 and 1+2+3 is the first part of the sum of the 
integers, from 1 to 5-2. So,

Sn = 4+2(n-1)*3+2(n-1)(n-2)/2.

Using this formula you can figure out the number of matches  needed 
for 6, 10, and 50 steps.  

By experimenting a little you can show that S71 = 5254.

- Doctor Jerry, The Math Forum   
Associated Topics:
High School Puzzles
High School Sequences, Series
Middle School Puzzles

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.