New Apartment Design

Last Friday we started work on the following problem:

We were amazed how many different patterns we found. Take a look at the list we came up with together:

Patterns we noticed

• Each time a row pushes up from the bottom with one extra block on the end, it’s like looking at a staircase from the side 
• In the chart it goes: 0+1=1, 1+2=3, 3+3=6, 6+4=10 (To get the total number of blocks, each time you add the number of the apartment to the previous total to get the new total. For example, in apartment #3 there were 6 blocks, to get apartment #4 you add 6+4=10.)
• The apartment number tells you the number of blocks on the bottom row and the left column
• The columns keep decreasing by one each time from left to right
• When the apartment number gets bigger, the actual building gets bigger too
• First it adds 1, then it adds 2, then it adds 3, then it adds 4, and so on.

Some of us are working on the challenge of finding how many blocks would be used in the 100th apartment. We know that drawing a picture of this apartment would take FOREVER, so we are we using what we know about the pattern to find the answer.