+0

toothpicks

+3
298
5

250 toothpicks how many rows (pyramid) could you make

Guest Jan 4, 2016

#3
+78620
+10

Very nice, Melody......in a slightly different manner.....the sum of the first n integers  =

[n (n + 1)] / 2  <=  250

n^2 + n <= 500

n^2 + n - 500 <=  0

On the graph here :  https://www.desmos.com/calculator/ajecyupfie , the largest positve root  = about 21.87....so,  we can take the floor of  21.87  as the answer   =  21 rows

CPhill  Jan 5, 2016
Sort:

#1
+2493
0

i think 250/8 = 31.25  (31)

Solveit  Jan 4, 2016
#2
+91024
+10

Hi Solveit and Guest ,

250 toothpicks how many rows (pyramid) could you make

1 on the top

2 on the next row

3 on the next row  etc

$$1+2+3+4+...... +n \le 250\\~\\$$

1+2+3+....+n

is the sum of an AP

$$a=1\;\;\;d=1 \;\;\;\\ Sn=\frac{n}{2}(a+L)\\ Sn=\frac{n}{2}(1+n )\\ so\\ \frac{n}{2}(1+n )\le250\\ n(1+n )\le500\\ n^2+n-500 \le0\\ \mbox{Solve for 0}\\ n=\frac{-1\pm\sqrt{1+2000}}{2}\\ n=\frac{-1\pm\sqrt{2001}}{2}\\ \mbox{n is not negative}\\ n=\frac{-1+\sqrt{2001}}{2}\\ n=21.8\\ \mbox{so there can be 21 rows}$$

231 toothpickswill be used andthere will be 19 left over :)

Melody  Jan 4, 2016
#3
+78620
+10

Very nice, Melody......in a slightly different manner.....the sum of the first n integers  =

[n (n + 1)] / 2  <=  250

n^2 + n <= 500

n^2 + n - 500 <=  0

On the graph here :  https://www.desmos.com/calculator/ajecyupfie , the largest positve root  = about 21.87....so,  we can take the floor of  21.87  as the answer   =  21 rows

CPhill  Jan 5, 2016
#4
+26328
+5

Suppose the toothpicks are arranged as follows (i.e. 3 per row; after all they are long and thin!):

Then you get 12 rows with 16 left over.

Alan  Jan 5, 2016
#5
+91024
0

That is great Alan.  I bet that is exactly what was intended.

I did think my pyramid was very weighed down (flattened).  Maybe with time.  LOL

Then again maybe my toothpicks were standing upright.  That would have made a very tall skinny 'tower' I guess. :)

Melody  Jan 5, 2016
edited by Guest  Jan 5, 2016

30 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details