# Finish this sequence of equalities…

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 = ?

Hint: My wife found it immediately. Me, with my Ph.D. in Mathematics, I could not find it.

Source: Misha Lemeshko ### Daniel Lemire

A computer science professor at the University of Quebec (TELUQ).

## 52 thoughts on “Finish this sequence of equalities…”

1. Mark Reid says:

Cute. That took me a minute or so before, thanks to your hint, I switched my maths brain off and saw it.

Those looking for another hint should think “topologically” (or perhaps homotopically).

2. Geoff Wozniak says:

I like that one. My wife didn’t get it, but I expect that’s because I bothered her when I introduced it to her.

3. Daniel Tunkelang says:

It’s the opposite of an open problem. 🙂

4. phd student says:

thanks God I don’t have a phd yet, I was able to solve it

5. Michael Lugo says:

4444 = ?

6. Karthik says:

2581 = 2?

7. Mark Reid says:

@Michael Lugo

4444 = 4 or 0, depending on which font you use.

8. David says:

2581 = 2 ?

9. Sui Fai John Mak says:

Here is the reason:
The number of 2 is the 21 number
If you subtract the 7th number (2) by 6th number (0)= 2
If you subtract the 14th number (5) by 13th number (3) = 2
If you subtract the 21st number (x)by 20th number (0), it should give you 2
I don’t have a PhD yet. But I am passionate in Mathematics, and is interested in Game Theory in my Operational Research studies in my MSc.
It still took me 5 minutes in working it out. I could see why it is difficult. Simply because people would like to share the problem, but how about the solution….that is outside the square!
You are welcome to visit my blog
http://suifaijohnmak.wordpress.com
I could make up this mathematical problems if you like.
John

10. Stephen Downes says:

2

11. David says:

2581 = 2

map 0, 6, and 9 to a value of ‘1’
map 8 to a value of ‘2’.
map any other digit to ‘0’

sum up mapped digit values of the left number and you obtain the number on the right.

examples:

8809 = 6: 8(2) + 8(2) + 0(1) + 9(1) = 6
2222 = 0: 2(0) + 2(0) + 2(0) + 2(0) = 0
6855 = 3: 6(1) + 8(2) + 5(0) + 5(0) = 0

therefore,

2581 = 2: 2(0) + 5(0) + 8(2) + 1(0) = 2

12. David says:

I challenged a friend with this problem and he answered: “just count the number of circles found on the left”.

I prefer his solution to mine.

13. Kevembuangga says:

Just a few words of rambling on that.
“Salience of artifacts upon a shifted representation of the problem”
The question remains, how do we (usually) spot the fruitful representation?

14. Paranoid says:

“My wife solved it” is much too great of a hint. Took about two seconds to solve it after I read that. 🙂

15. Paranoid says:

(am I sexist or just psychologically well-educated?)

16. Sammy says:

First Betti number of the collection of digits, thought of as a topological space!

17. man from Georgia says:

we must count balls.

consider 7662 here are 2 balls in 66

also

1111 here is no balls :))))))))

18. keios says:

Hi folks, so here’s my solution:

1. Let’s take only the result column
2. There, the zeros and their count does not count
3. 2 (other than 1) numbers are followed by their difference

So:
the difference of 6 & 4 is 2
the difference of 1 & 4 is 3
the difference of 5 & 4 is 1
the difference of 3 & 5 is 2

Cheers!

19. keios says:

Sorry: … 2 (other than 0)…

20. Maciek says:

Bartek 🙂 faktycznie

21. Bartek says:

Hi. All we need is to count the number of rings/circles number. 😀 For example in 0000 are 4, in 8888 are 8 and in 1111 – 0. Your wife found it, cause women see things differently. ^^

22. Crowell says:

0=4
1=0
2=0
3=0
4
5=0
6=4
7=0
8
9=4
and /4

sum of numbers for example 7756 = [7(0)+7(0)+5(0)+6(4)]/4=1
but this solution with circles.. wow. It took me 2 days (2-4h in sum) to figure it out

23. Crowell says:

sory, 8=8 🙂

24. killyox says:

There’s no maths here !
Just forms !
You would find easier the solution with paint than with a calculator.

Solution :
The result of each line is the number of “circle” in this line.
0 = 1 circle
8 = 2 circles (1 circle above 1 circle)
1 = 0 circle
2 = 0 circle
6 = 1 circle
9 = 1 circle
etc.

So 2581 :
2 = 0 circle
5 = 0 circle
8 = 2 circles
1 = 0 circle
=> Solution = 2

Thanks ! 🙂

25. EmmaBlue says:

Kylliox, you are a star. Thank you. I finally understand what is this discussion about:)

26. D says:

I could not find the answer and I was going crazy! I literally felt very, very uncomfortable..
I only understood after reading all the explanations twice..
I kept asking myself, what circle? what balls? what the hell! then I saw ‘no math’
whew… I finally understand..
balls, circles and no math

8 = 2 – one ball/circle/closed loop on top, one below the other.. put 2 circles one on top of the other and you’ll get 8

9 and 6 = 1 because there’s only 1 loop/ball/circle, when you write the number 9 or 6, the line touches itself..
or draw a circle, and add/join a curve below to get a 9,
or draw a circle, add/join a curve to the top and you’ll get 6

27. Anonymous says:

What i did was just define the numbers I need as variables by looking at the values listed. Simply add them up and done. 0=1, 1=0, 2=0, 3=0, 5=0, 6=1, 7=0, 8=2, 9=1 and I believe 4 wasn’t listed. amirite? So for 2581: 2=0,5=0,8=2 and 1=0. The total would be 2.

28. William Wu says:

It is 2012 now…
IT Person find the result within 2 minutes….by Goggle

29. David says:

There is a ‘mathematical’ way to solve this.

Sort the rows from highest result to lowest.

The typical pattern present on the top will serve as a good hint to what exactly is counted here.

30. terrypin says:

It’s now Saturday 3 March 2012.

The Facebook post I saw this morning
from my son included the hint that pre-school children could solve it in 5 minutes. So it took me only a minute or two. I’d still be staring at it otherwise!

His version didn’t include the line
2581 =
so I was just looking for the ‘rule’.

But if the last line had been
2854 =
then it complicates things.
SPOILER

If the rule is ‘number of circles’ the answer remains 2. But if it’s ‘holes you can fill with crayon or paint’ then it’s 3.

Terry, UK

31. Cindy says:

Haha, it’s funny that people are mapping numbers and thinking of crazy two day algrms, when the answer is not that at all!

32. Mihai says:

This is stupid, really :))

33. johnpaul says:

Simply count the closed loops…
8=2 closed loops
9=1 closed loop
0=1 closed loop
all other numbers no loop

34. Andrew says:

Or you could simply use a polynomial that is able to give you the above results… any mathematician worth his salt should be able to get to this answer.

35. Daniel Lemire says:

@Andrew

Solving for a polynomial of degree 19 is not the sought-after solution.

36. Smola says:

So, similar problem can be made with horizontal bar count (2457=>4), or not circles, or left curles and horizontal bars but only if the first one is a circle, or… 🙁

37. Sundar says:

Hi Friends

1. Take a count of numbers each number is divisible by excluding the number itself and 1
2. For ‘0’ , 0 is always divided by any number so its 1

2581 = 2

For

1. 2 can be divided by 1 and 2
But Apply my rule 1
So Count is 0
2. 5 can be divided by 1 and 5
Apply rule 1
So Count is 0
3. 8 can be divided by 1, 2, 4 and 8
Apply rule 1
So count is 2
4. 1 is divisible by 1
Apply rule 1
So count is 0

Add the total count, correct me if I am wrong

– Sundar –

38. romadhoni says:

you calculate it for too long and too complicated.
see all the numbers there must be a hole or circle. Suppose 0 circle 1, 6 circle 1, circle 2 8, 9 circle 1. So if 0000 = 4. If 2411 = 0, because there is no circle on the numbers. So after this you will know the answer to that question.

39. Tommaso says:

Sundar, according to your rule I don’t see how 6666=4 :
6 can be divided by 1, 2, 3 and 6
Apply rule 1
So Count is 2
6666 should be 8 and it isn’t…
So, in my opinion, children solution of counting the circles is still the only solution…

40. irish joy PHD says:

OMG.. this is so easy and youre having difficult times with..i found the answer for only 2 mins, and it counts the number of holes in the numbers…lol.. such a preschoolers mind riddle

41. Christie says:

I love how simple this was to solve, once you knew what to look for, and how interesting it was that so many people came up with math solutions to solve the problem doing mathmatics no preschooler could ever understand. How far we have gone from simplicity.

42. thereFOUR says:

I gave up after thinking 2 hours and read the comments. I feel stupid.

43. Rodrigo says:

“2581=?”
2581=2
8 tiene dos circulos 8=2
2,5 y 1 no tienen circulos por lo tanto
2=0 5=0 1=0 circulos

44. HJ says:

Crowell- I was fascinated by your answer. How did you get to it?
But then I realized: Why didn’t you just multiply all the values you gave by 10, and then, instead of dividing by 4, divide by 40??
Or, more simply, why did you bother giving 8 a value of 8 and 0 a value of 4, etc. and then having to divide by 4? Just give each number the value of its number of circles, and then you won’t have to divide at all!!

45. TB says:

I got the answer to be 3, so I would say there is not enough information to come to the author’s conclusion. I used a simple mapping of digits and added them. All the answers are correct.

0, 6, 9 = 1
1, 2, 3, 4, 5, 7 = 0
8 = 2

Therefore, 2581 = 3

46. å­Ÿæµ·æ¸¯ says:

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47. å­Ÿæµ·æ¸¯ says:

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48. J.C. says:

The only reason I figured this out was because my preschoolers never did math until 1st grade! Once I stopped thinking math, I saw the answer right away!

49. Anonymous says:

8: 8888=1#
7: 6,9,0-888=3×4=12#
6: 1,2,3,4,5,7-888=6×4=24
6,9,0-88=18+36=54
#24+54=78#
5: 1,2,3,4,5,7-6,9,0-88=
18×12=216
6,9,0-8=12+72+24=108
#216+108=324#
4: 1,2,3,4,5,7-88=36+180
=216
1,2,3,4,5,7-6,9,0-8=
216+(18×24)=648
6,9,0=3+24+18+36=81
#216+648+81=945#
3: 1,2,3,4,5,7-6,9,0-8=(18×12)+
(15x3x24)=216+1080=1296
1,2,3,4,5,7,-6,9,0=
72+432+144=648
#1296+648=1944#
2: 1,2,3,4,5,7-8=24+360+480=864
1,2,3,4,5,7-6,9,0=
108+216+540+1080=1944
#864+1944=2808#
1: 1,2,3,4,5,7-6,9,0=
#72+1080+1440=2592#
0: 1,2,3,4,5,7=
#6+120+90+720+360=1296#

0-8 ttl: 1+12+78+324+945+1944+2808+
2592+1296=10000

Gary Wong

50. gary wong says:

8: 8888=1#
7: 6,9,0-888=3×4=12#
6: 1,2,3,4,5,7-888=6×4=24
6,9,0-88=18+36=54
#24+54=78#
5: 1,2,3,4,5,7-6,9,0-88=
18×12=216
6,9,0-8=12+72+24=108
#216+108=324#
4: 1,2,3,4,5,7-88=36+180
=216
1,2,3,4,5,7-6,9,0-8=
216+(18×24)=648
6,9,0=3+24+18+36=81
#216+648+81=945#
3: 1,2,3,4,5,7-6,9,0-8=(18×12)+
(15x3x24)=216+1080=1296
1,2,3,4,5,7,-6,9,0=
72+432+144=648
#1296+648=1944#
2: 1,2,3,4,5,7-8=24+360+480=864
1,2,3,4,5,7-6,9,0=
108+216+540+1080=1944
#864+1944=2808#
1: 1,2,3,4,5,7-6,9,0=
#72+1080+1440=2592#
0: 1,2,3,4,5,7=
#6+120+90+720+360=1296#

0-8 ttl: 1+12+78+324+945+1944+2808+
2592+1296=10000

Gary Wong

51. gary wong says:

by this logic, count from 0000 to 9999,the total no of 0,1,2,….7,8 are:

8: 8888=1#

7: 6,9,0-888=3×4=12#

6: 1,2,3,4,5,7-888=6×4=24
6,9,0-88=18+36=54
#24+54=78#

5: 1,2,3,4,5,7-6,9,0-88=18×12=216
6,9,0-8=12+72+24=108
#216+108=324#

4: 1,2,3,4,5,7-88=36+180=216
1,2,3,4,5,7-6,9,0-8=216+(18×24)=648
6,9,0=3+24+18+36=81
#216+648+81=945#

3: 1,2,3,4,5,7-6,9,0-8=(18×12)+(15x3x24)=1296
1,2,3,4,5,7,-6,9,0=72+432+144=648
#1296+648=1944#

2: 1,2,3,4,5,7-8=24+360+480=864
1,2,3,4,5,7-6,9,0=
108+216+540+1080=1944
#864+1944=2808#

1: 1,2,3,4,5,7-6,9,0=
#72+1080+1440=2592#

0: 1,2,3,4,5,7=
#6+120+90+720+360=1296#

0-8 total: 1+12+78+324+945+1944+2808+
2592+1296=10000

Gary Wong

52. boo xi says:

the rule of a circle is the best rule, so its the only way out

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