Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2016-06-26 15:22:01

Mathegocart
Member
Registered: 2012-04-29
Posts: 2,226

Dots and toothpicks

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.
3e647e404aae07454e84e5e36b2f00ba2f4ba047.png
(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?
Me: What I tried to do was find a pattern:
a 2 x 2 square has 12 toothpicks. 4 of the "sides" are shared and 8 are not shared. A 1 vert x 2 horiz  has 1 shared side and 6 n.s sides(7). A 3 x 3 square has 8 shared sides and 12 unshared sides. 20. I gave up after this
(b) Can you generalize your answer? Suppose we have a grid that requires $h$ horizontal toothpicks in each row and $v$ vertical toothpicks in each column. Then, how many total toothpicks will we need? Also, how many total dots are there?
???? I have no clue on how to start this?


The integral of hope is reality.
May bobbym have a wonderful time in the pearly gates of heaven.
He will be sorely missed.

Offline

#2 2016-06-26 16:35:33

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: Dots and toothpicks


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

Offline

#3 2016-06-26 19:01:57

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Dots and toothpicks

Hi;

I have no clue on how to start this?

Here is one way to start.

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?

Play spot the pattern.

o0qz95M.png

We will have the computer count those toothpicks for us.

{7, 22, 45, 76, 115, 162, 217...}

It is easy to come up with an equation for this:

where n is the number of horizontal toothpicks and 2n is the number of vertical ones. So, for a 10 toothpick across we have 30 + 400 = 430

Here is a nice recurrence:

with a(0) = 0, a(1) = 7, a(2) = 22

Here is a simpler recurrence:

with a(0) = 0.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

Board footer

Powered by FluxBB