How To Really Solve The Straight Lines And Triangles Puzzle?
Published on April 25, 2022 by Hemanth
--
The straight lines and triangles puzzle follows directly from my essay on the quadratic equation magic trick. In this puzzle, your job is to figure out how many triangles you can construct using ’n’ straight lines.
The following are the conditions of the puzzle:
1. No two lines shall be parallel to each other.
2. Each line shall intersect every other line.
3. Any intersection point shall involve no more than two straight lines.
Given these conditions, how would you proceed? As a hint, you may look for a clue in the essay that I linked at the beginning.
Spoiler Alert:
If you wish to solve this puzzle on your own without any further clues, now is a good time to tune off of this essay. Once you are done, you may choose to continue reading this essay. Beyond this section, I will be discussing direct solutions to this puzzle.
Let us start with the classic trial and error approach. But even before that, using geometric logic alone, we could establish that we can construct no triangle using less than three straight lines. So, the solution that we are looking for is likely an expression that yields zero for n = 0,1, and 2 respectively.
Let us now see what happens when we have three straight lines that follow the conditions of the puzzle:
Illustration created by the author
As we can clearly see, this leads to a situation where we can construct one solitary triangle. So, the expression we are looking for yields the result of 1 for n = 1.
Let us now see what happens when we have four lines:
Illustration created by the author
We see from the illustration that we can construct a total of four lines under the conditions of the puzzle. Let us now take it one step further and see what the problem space with five straight lines looks like:
Illustration created by the author
It took a lot more effort and it looks a lot messier. However, we do have an answer in the end. We can construct a total of 10 triangles using 5 straight lines under the given conditions.
We could keep going, but not only does it get significantly more cumbersome, but now we have sufficient data to try and workout the underlying expression.
Using Calculus of Finite Differences to Solve the Straight Lines and Triangles Puzzle
As I had hinted at the beginning of the essay, we are going to employ the calculus of finite differences to workout the underlying expression. Let us first write down the results we have and construct the inverted pyramid:
Math illustrated by the author
We have ended up with alike numbers in the third row of differences. What this means is that the expression we are looking for is cubic.
Likewise, if we had ended up with alike numbers in the second row of differences, the expression would be quadratic. And if it had been in the first row of differences, our expression would have been linear. In short, the number of rows of differences is equal to the order of the expression we are looking for.
The Genius of Newton
Luckily, for the method we are pursuing, Isaac Newton had come up with a formula that works well. If we express the value of the expression for n = 0 as ‘a’, the first number in the first row of differences as ‘b’, the first number in the second row of differences as ‘c’, and so on, Newton’s formula is as follows:
Math illustrated by the author
If we plug in the values we have from our inverted pyramid, we end up with the following expression:
Math illustrated by the author
If you look at this closely, this expression is none other than the value of nC3 (where C is the combination function from combinatorics). This is because our geometric puzzle could be essentially transformed as the following question:
How many combinations of groups of three straight lines from a total of ’n’ straight lines are possible?
If it was so simple to transform the puzzle into this question and get the answer using combinatorics, why did I not reveal it until this point? The reason is that this puzzle serves as a good step into the world of the calculus of finite differences.
Having come this far, let us take a look at some of the finer properties of the method of finite differences.
A Finer Look into the Method of Finite Differences
Here is a recap of how I solved the puzzle:
1. I used trial and error to get the pattern of the outputs for inputs from an arithmetic series.
2. I then used the method of finite differences to work out the expression that fits the observed output.
One point to note about this approach is that even though this is very effective, my solution is still just a “guess”. Until I prove that the pattern holds indefinitely, this remains the case.
In short, the method of differences is only as good as the validity of the pattern observed. This presents problems when we try to apply this method to real-world problems. We often end up with expressions that are true until they are not.
Often, no finite amount of data can sufficiently prove the validity of a pattern. And just one data point is sufficient to disprove the pattern observed until that point. As mathematician George Pólya put it:
“Nature may answer Yes or No, but it whispers one answer and thunders the other.”
Another point to note about this method is that it works only for polynomial expressions. Had our original expression been exponential, the consequent rows of differences would lead to recursive spirals.
Final Thoughts
Westarted with straight lines and triangles and ended up stepping into the world of the calculus of finite differences. Along the way, we covered some of its finer properties and limitations.
Even considering the limitations, the method of differences happens to be used in studying real-world phenomena, numerical algorithms, and iterative mathematical schemes, among others.
Now that you have had exposure to this method, what do you think you could use it for?
We use cookies on our website to give you the most relevant experience by remembering your preferences and repeat visits. By clicking “Accept”, you consent to the use of ALL the cookies.
This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the ...
Necessary cookies are absolutely essential for the website to function properly. These cookies ensure basic functionalities and security features of the website, anonymously.
Cookie
Duration
Description
cookielawinfo-checkbox-advertisement
1 year
Set by the GDPR Cookie Consent plugin, this cookie is used to record the user consent for the cookies in the "Advertisement" category .
cookielawinfo-checkbox-analytics
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Analytics".
cookielawinfo-checkbox-functional
11 months
The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional".
cookielawinfo-checkbox-necessary
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookies is used to store the user consent for the cookies in the category "Necessary".
cookielawinfo-checkbox-others
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Other.
cookielawinfo-checkbox-performance
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Performance".
CookieLawInfoConsent
1 year
Records the default button state of the corresponding category & the status of CCPA. It works only in coordination with the primary cookie.
viewed_cookie_policy
11 months
The cookie is set by the GDPR Cookie Consent plugin and is used to store whether or not user has consented to the use of cookies. It does not store any personal data.
Functional cookies help to perform certain functionalities like sharing the content of the website on social media platforms, collect feedbacks, and other third-party features.
Performance cookies are used to understand and analyze the key performance indexes of the website which helps in delivering a better user experience for the visitors.
Cookie
Duration
Description
_gat
1 minute
This cookie is installed by Google Universal Analytics to restrain request rate and thus limit the collection of data on high traffic sites.
Analytical cookies are used to understand how visitors interact with the website. These cookies help provide information on metrics the number of visitors, bounce rate, traffic source, etc.
Cookie
Duration
Description
__gads
1 year 24 days
The __gads cookie, set by Google, is stored under DoubleClick domain and tracks the number of times users see an advert, measures the success of the campaign and calculates its revenue. This cookie can only be read from the domain they are set on and will not track any data while browsing through other sites.
_ga
2 years
The _ga cookie, installed by Google Analytics, calculates visitor, session and campaign data and also keeps track of site usage for the site's analytics report. The cookie stores information anonymously and assigns a randomly generated number to recognize unique visitors.
_ga_R5WSNS3HKS
2 years
This cookie is installed by Google Analytics.
_gat_gtag_UA_131795354_1
1 minute
Set by Google to distinguish users.
_gid
1 day
Installed by Google Analytics, _gid cookie stores information on how visitors use a website, while also creating an analytics report of the website's performance. Some of the data that are collected include the number of visitors, their source, and the pages they visit anonymously.
CONSENT
2 years
YouTube sets this cookie via embedded youtube-videos and registers anonymous statistical data.
Advertisement cookies are used to provide visitors with relevant ads and marketing campaigns. These cookies track visitors across websites and collect information to provide customized ads.
Cookie
Duration
Description
IDE
1 year 24 days
Google DoubleClick IDE cookies are used to store information about how the user uses the website to present them with relevant ads and according to the user profile.
test_cookie
15 minutes
The test_cookie is set by doubleclick.net and is used to determine if the user's browser supports cookies.
VISITOR_INFO1_LIVE
5 months 27 days
A cookie set by YouTube to measure bandwidth that determines whether the user gets the new or old player interface.
YSC
session
YSC cookie is set by Youtube and is used to track the views of embedded videos on Youtube pages.
yt-remote-connected-devices
never
YouTube sets this cookie to store the video preferences of the user using embedded YouTube video.
yt-remote-device-id
never
YouTube sets this cookie to store the video preferences of the user using embedded YouTube video.
Comments