We connect dots with toothpicks in a grid as shown below. For example, the grid below has 7 horizontal toothpicks in each row and 5 vertical toothpicks in each column.

(a) Suppose we instead have a grid of dots that requires 10 horizontal toothpicks in each row and 20 vertical ones in each column. Then, how many total toothpicks will we need? Also, how many total dots are there?

(b) Can you generalize your answer? Suppose we have a grid that requires horizontal toothpicks in each row and vertical toothpicks in each column. Then, how many total toothpicks will we need? Also, how many total dots are there?

Guest Feb 21, 2015

#2**+10 **

In each row there is one more dot than there are horizontal toothpicks, and in each column there is one more dot than there are vertical toothpicks, so:

a) Number of horizontal toothpicks = 10*(20 + 1) = 210

Number of vertical toothpicks = 20*(10 + 1) = 220

Total number of toothpicks = 430

Number of dots = (10 + 1)*(20 + 1) = 11*21 = 231

b) Number of horizontal toothpicks = h*(v + 1)

Number of vertical toothpicks = v*(h + 1)

... I'll leave you to finish off.

.

Alan
Feb 21, 2015

#1**0 **

Please explain thoroughly in this question. I am so sorry! I forgot to include this in the quesiton!! Thanks, this website is very helpful.

Guest Feb 21, 2015

#2**+10 **

Best Answer

In each row there is one more dot than there are horizontal toothpicks, and in each column there is one more dot than there are vertical toothpicks, so:

a) Number of horizontal toothpicks = 10*(20 + 1) = 210

Number of vertical toothpicks = 20*(10 + 1) = 220

Total number of toothpicks = 430

Number of dots = (10 + 1)*(20 + 1) = 11*21 = 231

b) Number of horizontal toothpicks = h*(v + 1)

Number of vertical toothpicks = v*(h + 1)

... I'll leave you to finish off.

.

Alan
Feb 21, 2015