# 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

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).

Comment by Mark Reid — 10/1/2009 @ 18:08

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.

Comment by Geoff Wozniak — 10/1/2009 @ 18:16

It’s the opposite of an open problem.

Comment by Daniel Tunkelang — 10/1/2009 @ 18:24

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

Comment by phd student — 10/1/2009 @ 18:36

4444 = ?

Comment by Michael Lugo — 10/1/2009 @ 18:43

2581 = 2?

Comment by Karthik — 10/1/2009 @ 21:00

@Michael Lugo

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

Comment by Mark Reid — 11/1/2009 @ 2:44

2581 = 2 ?

Comment by David — 11/1/2009 @ 8:56

The answer is 2.

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

Comment by Sui Fai John Mak — 11/1/2009 @ 18:39

2

Comment by Stephen Downes — 11/1/2009 @ 19:35

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

Comment by David — 11/1/2009 @ 20:31

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.

Comment by David — 12/1/2009 @ 12:42

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?

Comment by Kevembuangga — 13/1/2009 @ 4:23

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

Comment by Paranoid — 13/1/2009 @ 20:35

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

Comment by Paranoid — 13/1/2009 @ 20:36

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

Comment by Sammy — 1/3/2009 @ 19:17

we must count balls.

consider 7662 here are 2 balls in 66

also

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

Comment by man from Georgia — 9/7/2011 @ 14:54

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!

Comment by keios — 9/2/2012 @ 6:12

Sorry: … 2 (other than 0)…

Comment by keios — 9/2/2012 @ 6:13

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. ^^

Comment by Bartek — 11/2/2012 @ 15:10

Bartek faktycznie

Comment by Maciek — 11/2/2012 @ 19:22

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

Comment by Crowell — 12/2/2012 @ 16:40

sory, 8=8

Comment by Crowell — 12/2/2012 @ 16:41

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 !

Comment by killyox — 13/2/2012 @ 15:35

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

Comment by EmmaBlue — 16/2/2012 @ 11:30

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

Comment by D — 19/2/2012 @ 10:44

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.

Comment by Anonymous — 22/2/2012 @ 12:18

It is 2012 now…

IT Person find the result within 2 minutes….by Goggle

IT Youngsters find the result within 3 seconds…by Facebook, twitter….

Comment by William Wu — 24/2/2012 @ 4:01

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.

Comment by David — 28/2/2012 @ 12:23

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

Comment by terrypin — 3/3/2012 @ 6:52

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

Comment by Cindy — 6/3/2012 @ 23:14

This is stupid, really )

Comment by Mihai — 16/3/2012 @ 14:33

Simply count the closed loops…

8=2 closed loops

9=1 closed loop

0=1 closed loop

all other numbers no loop

Comment by johnpaul — 17/3/2012 @ 8:06

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.

Comment by Andrew — 20/3/2012 @ 4:37

@Andrew

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

Comment by Daniel Lemire — 20/3/2012 @ 7:34

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…

Comment by Smola — 20/3/2012 @ 18:12

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 -

Comment by Sundar — 21/3/2012 @ 16:01

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.

Comment by romadhoni — 29/3/2012 @ 0:27

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…

Comment by Tommaso — 30/3/2012 @ 14:41

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

Comment by irish joy PHD — 7/4/2012 @ 9:00

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.

Comment by Christie — 18/4/2012 @ 23:03

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

Comment by thereFOUR — 28/4/2012 @ 4:55

“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

Comment by Rodrigo — 1/5/2012 @ 23:22

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!!

Comment by HJ — 6/5/2012 @ 9:27

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

Comment by TB — 12/5/2012 @ 23:41

当然是2啦 有几个圈就是几

Comment by 孟海港 — 18/5/2012 @ 0:08

当然是2啦 有几个圈就是几

is 2。how many circles ，it is the num。

用不着算法~ 哈哈

Comment by 孟海港 — 18/5/2012 @ 0:12

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!

Comment by J.C. — 3/6/2012 @ 14:21

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

Comment by Anonymous — 29/11/2012 @ 3:59

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

Comment by gary wong — 29/11/2012 @ 4:01

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

Comment by gary wong — 29/11/2012 @ 4:12