## An answer to a puzzle

The puzzle I posted a few days ago is derived from a puzzle that’s worked its way around the Internet every so often in the past few years. See this forum in Russian (why don’t I know Russian?), or Misha Lemeshko’s blog, or Daniel Lemire for the version that mine is derived from. The incarnation I saw on Wednesday, on Facebook, which inspired this post, says that “This problem can be solved by pre-school children in 5-10 minutes, by programmers – in 1 hour, by people with higher education… well, check it yourself! :)

It’s then followed by the following list of numbers:

8809 = 6
7111 = 0
2172 = 0
6666 = 4
1111 = 0
3213 = 0
7662 = 2
9312 = 1
0000 = 4
2222 = 0
3333 = 0
5555 = 0
8193 = 3
8096 = 5
7777 = 0
9999 = 4
7756 = 1
6855 = 3
9881 = 5
5531 = 0

2581 = ?

Sort of implicit in the hint is that maybe it has something to do with the digits; Real Mathematicians think that puzzles involving digits are somehow inferior. (In A Mathematician’s Apology Hardy observes of facts such as 8712 = 4 × 2178 and 153 = 13 + 5 3 + 33 that “[t]hese are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals much to a mathematician.”

Perhaps the natural thing to do, if this is a claim about digits, is to then assume that the claim 8809 = 6 encodes the statement $f(8) + f(8) + f(0) + f(9) = 6$, and proceed on that basis. On that basis we have $4f(1) = 0, 4f(0) = 4, 4f(2) = 0, 4f(3) = 0, 4f(5) = 0, 4f(6) = 0, 4f(7) = 0, 4f(9) = 4$ and so $f(0)=f(9) = 1, f(1)=f(2)=f(3)=f(5)=f(6)=f(7)= 0$. We still need $f(4), f(8)$. From the first “equality” we have $2f(8) + f(0) + f(9) = 6$ and so $f(8) = 2$. The answer is $f(2) + f(5) + f(8) + f(1) = 0+0+2+0 = 2$.

As for the version I gave — there are eight equations in nine unknowns. These were derived by removing from the “bloated” version of the puzzle all the equations with four of the same digits on the left side, and all those with a zero on the right side. The system again has equations $f(8) + f(8) + f(0) + f(9) = 6$ and so on. By subtracting equations from each other we get $f(0) = 1 + f(1), f(3) = f(2), f(6) = 1+f(2), f(8) = 2+f(2)$. From “6855 = 3″ we have $f(6) + f(8) + 2f(5) = 3$, or $2f(2) + 2f(5) = 0$; if we agree that all values are nonnegative then $f(2) = f(5) = 0$. Then from “7756 = 1″ we can get $f(7) = 0$. Also from relations we alrady derived, $f(3) = 0, f(6) = 1, f(8) = 2$.

But what are the values of 0, 1, and 9? It turns out that either $(f(0), f(1), f(9)) = (1,0,1)$ or $(2,1,0)$ works, from an addition standpoint.

So why does 4 never appears on the left hand side, therefore meaning we can never work out $f(4)$. This is a feature, not a bug. $f(n)$ is the number of holes in the numeral n. Some people draw 4 with one hole; some draw it with zero. So we choose $(f(0),f(1),f(9)) = (1,0,1)$ and so the answer is $f(2) + f(5) + f(8) + f(1) = 0 + 0 + 2 + 0 = 2$.

About these ads
This entry was posted in Uncategorized. Bookmark the permalink.

### 38 Responses to An answer to a puzzle

1. jc says:

For what it’s worth, I didn’t get the puzzle at all until after I saw the hint that pre-schoolers could solve it in 5-10 minutes, and then while scrolling through the list the key jumped out at me immediately even before I got to the rest of your post.

2. Just a side note as someone who’s used LaTeX on WordPress for a while: the {aligned} environment is your best friend for lists of equations like

\begin{aligned}4f(1)=0\\4f(0)=4\\4f(2)=0\\4f(3)=0\\4f(5)=0\\4f(6)=0\\4f(7)=0\\4f(9)=4\end{aligned}

3. ae says:

You used 6666=0 in your solution. The puzzle says 6666=4.

4. Pep95 says:

Just count the circles

5. Pep95 says:

Like 6 has one circle, so 6=1
1 has no circles, so 1=0
8 has two circles, so 8=2
etc.

6. Ahmed Atef says:

just count the number of circles of each No. in each series and you’ll get the answer
i.e 6666=4 circles
8809=6 circles
7111=0 circles

7. kingkong says:

Why would kids look for circles in particular? I might have looked for any other shape. This is nonsense. *Damn, those grapes look sour*

8. nice going says:

dude you are so retarded, just write what the values are and add them… dumbasses

9. nice going says:

king kong you genius

10. Wayne says:

The answer is 2. Just count the number of circles in these numbers… for example, 9 has 1 circle and 8 has 2, 0 has 1, and 7 has no circle….

11. amit khan says:

2

12. Gene says:

It’s an awesome post in support of all the online visitors; they will obtain advantage from it I am sure.

13. Luke says:

My brother just emailed me this, and I found the answer to be 2 also, but had a different reason. I found that numbers that could be rotated 180 degrees and still be numbers are worth 1 and numbers that can be rotated to a side and still be numbers (or mathematical anyway) are worth 2. 8 is the only number worth 2, because it is viewed as the infinity symbol sideways.

14. Luke says:

I just found a third way, I think. When drawing the number, without lifting the pen, the minimum number of times that the pen comes in contact with itself by completion of the number, without lifting the pen and without retracing a part of the number is the value of the number. 8 must contact itself twice when drawn without lifting. 0, 6, 9 must contact once. all other numbers do not contact themselves without retracing.

15. Luke says:

I forgot 4, which doesn’t appear, but would also equal 1 by this third rule,

16. Luke says:

I think 4 is quite clearly, intentionally, left out of the left side of the puzzle. By the circle rule, 4 = 0. By a “closed loop” rule, 4 =1. By a rotational rule, 4 = 0. By a “minimum point of contact when drawing, without retracing” rule, 4=1. Alright, time for sleep.

17. On a plus musical note, Apple is functional on a local itemization armed
service and we distaff escorts Conduct you to disco biscuit.

18. I blog frequently and I seriously appreciate your information.
This article has truly peaked my interest. I am
going to book mark your site and keep checking for new information about once
per week. I subscribed to your RSS feed too.

19. smoke assist says:

It’s really a nice and helpful piece of info. I’m satisfied that you just shared this useful info with us.

Please stay us informed like this. Thank you for sharing.

20. Stormy says:

Hi friends, its impressive article regarding cultureand completely defined, keep
it up all the time.

21. I comment when I like a article on a website or if I have
something to contribute to the conversation.

Usually it’s a result of the passion communicated in the post I read. And on this post An answer to a puzzle | God plays dice. I was moved enough to post a comment :-) I actually do have some questions for you if it’s allright.
Is it only me or do a few of the comments look like coming from brain dead people?
:-P And, if you are writing at additional online sites, I would like to follow
anything fresh you have to post. Would you make
a list all of your shared pages like your twitter feed,
Facebook page or linkedin profile?

22. It’s actually a cool and helpful piece of information. I am glad that you shared this useful info with us. Please keep us up to date like this. Thanks for sharing.

23. Good information. Lucky me I recently found your website by chance (stumbleupon).
I’ve book-marked it for later!

24. Your style is very unique in comparison to other people I’ve read stuff from.
Many thanks for posting when you have the opportunity, Guess I
will just bookmark this web site.

25. Youtube.Com says:

My programmer is trying to convince me to move to .net from PHP.
I have always disliked the idea because of the expenses. But he’s tryiong none
the less. I’ve been using Movable-type on a variety of websites for about a year and am concerned about switching
to another platform. I have heard very good things about blogengine.net.
Is there a way I can transfer all my wordpress posts into it?
Any kind of help would be really appreciated!

26. I think this is among the such a lot significant info for me.
And i’m happy reading your article. But should observation on some basic things, The web site style
is ideal, the articles is actually nice : D. Excellent process, cheers

27. Hello colleagues, how is the whole thing, and what you want to say about this article, in my view its genuinely remarkable in support
of me.

28. First off I want to say great blog! I had a quick question that I’d like to ask if you do
not mind. I was curious to know how you center
yourself and clear your head before writing.

I have had trouble clearing my mind in getting my
thoughts out. I do enjoy writing but it just seems like the first 10 to 15 minutes are lost simply just trying
to figure out how to begin. Any recommendations or hints?
Many thanks!

29. 雨の音してます。幸せな素敵な恋愛がしたいですね。 裏事情もそれなりに知っているが、裏では金づるとしか思ってないよ。
カルビ好きを2回もアピールしてるとこみると いや～一万円でも、出してくれるだけで良いと思わないとダメ。 結婚祝い相場友人

何時間も拘束されてマズい料理食わされてしょうもない引き出物持たされて、県内だと足代もかかるクソイベントに行きたい奴なんていねーから えつさん(女性 39歳 O型
東京都)2014年05月15日(木) 22時41分 結婚祝い相場友人
最後にメールした翌日に事故にあって二週間ほど既読スルー状態でメール出来なかったんだけど、 披露宴によばれたら例え気持ちはなくともお金は出さんとあかん。

30. Most internet marketers try tto use SEO, forum marketing, article marketing and other forms of online marketing
which has mad the world of online marketing very packed out and hard to taap into.

Think of your audience as only aving fifteen minutews to spend on each business plan that comes across their desks.
Before setting out to buy a boox truck, youu should analyze what type and size you need.

31. You need to hold off a severe conversation you are planning to have with
somebody. Each time you inhale, this expands, with the help of the universe’s all-natural protecting energies.
What have their motives and motivations been recently?

32. google says:

All such apparatus can be easily maintained via the Google Apps backup tool.
Google Nexus 7 is definitely the companion you need. After you run your search, you’ll find the PPC ads on the
right-hand side of the screen.

33. If you wish for to increase your know-how only keep visiting this website
and be updated with the latest news posted here.

34. Hi there colleagues, its wonderful piece of writing about teachingand
fully defined, keep it up all the time.

35. curry goat says:

More than 20 million turkeys are produced for meat in the UK each
year that will earn their producers. As Horlicks contains all the essential nutrients, most parents choose to make it a
part of their children’s regular diet. Why should you make a natural skin moisturizer for toddler body care.

36. What i don’t realize is in fact how you’re no longer really much more neatly-appreciated than you may be right now.

You’re so intelligent. You recognize therefore considerably with regards to
this topic, made me for my part believe it from a lot of numerous angles.

Its like men and women are not interested except it’s something to do with Girl
gaga! Your personal stuffs excellent. At all times take care of it up!

37. Azmat Azzam says:

2581 = 2 it is a correct Answer….. :)