### Brain Teasers

# Monk Steps

A monk has a very specific ritual for climbing up the steps to the temple. First he climbs up to the middle step and meditates for 1 minute. Then he climbs up 8 steps and faces east until he hears a bird singing. Then he walks down 12 steps and picks up a pebble. He takes one step up and tosses the pebble over his left shoulder. Now, he walks up the remaining steps three at a time which only takes him 9 paces. How many steps are there?

### Answer

There are 49 steps.He climbs halfway, which is step 25. He hears the bird singing on step 33. He picks up the pebble on the 21st step and tosses it on the 22nd step. The remaining 27 steps are taken three at a time which is 9 paces.

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## Comments

It is not very hard : you just have to visualize the steps to find the answer, but I liked it. It is a fun one.

It's not hard after you have been told the answer to verify that it works. How do you work it out?

The key is working backwards from the top step.

If he covers 27 (3x9) steps to reach the top he must have been 28 steps from the top. Thus before he stepped up one he was 29 from the top. Before he went down 12 he was 17 from the top. Before he went up 8 he was 25 from the top. This is the middle step - 24 above it and 24 below it = 49 steps.

The key is working backwards from the top step.

If he covers 27 (3x9) steps to reach the top he must have been 28 steps from the top. Thus before he stepped up one he was 29 from the top. Before he went down 12 he was 17 from the top. Before he went up 8 he was 25 from the top. This is the middle step - 24 above it and 24 below it = 49 steps.

Yeah, I worked backwards to solve it. If you work forwards, it gets sort of complicated and confusing. Very well done.

Got it! Great teaser!

I liked it!

Vry tidy!

Vry tidy!

good one...but too much math in it

I just figured he was on the middle step, then goin up 8 and down 12 puts him at 4 below middle. Then one step up. Then one set of three put's him back on the middle. So there's 24 ahead of him, and 24 behind. 49. Many ways to solve this teaser. (or, many algorithms) Anyway, good teaser!

i didn't get it. too hard for me.

great math teaser.I liked it.

found a flaw...it says "he climbs up to the middle step" ...if there are 49 steps then the middle one would be the 24.5th step...

There are 48 steps. The middle step is the 24th.

Look at it this way: if there are two steps, the second step is the top level, the 1st step is in the middle. There has to be an even number of steps from the starting level to the top level, for there to be a middle step.

Look at it this way: if there are two steps, the second step is the top level, the 1st step is in the middle. There has to be an even number of steps from the starting level to the top level, for there to be a middle step.

Actually no there has to be an odd number of step to have a middle step, and the 25 would be the middle, so check your facts before trying to prove people wrong

I counted it with this equation(mentally o course )

x = 0,5x + 0,5 + 8 - 12 + 1 + 3*9

Because there is a middle step, the number of steps is odd. 0,5x + 0,5 calculates the middle step

thus, examples:

total steps: 25

middle step: 0,5*25 + 0,5 = 13

total steps: 49

middle step: 0,5*49 + 0,5 = 25

x = 0,5x + 0,5 + 8 - 12 + 1 + 3*9

0,5x = 24,5

x = 49

x = 0,5x + 0,5 + 8 - 12 + 1 + 3*9

Because there is a middle step, the number of steps is odd. 0,5x + 0,5 calculates the middle step

thus, examples:

total steps: 25

middle step: 0,5*25 + 0,5 = 13

total steps: 49

middle step: 0,5*49 + 0,5 = 25

x = 0,5x + 0,5 + 8 - 12 + 1 + 3*9

0,5x = 24,5

x = 49

Oh, and there was supposed to be spaces between lines in the above post, try to undestand

Impressive teaser ( i came up with 50 ) and impressive comment ched4r

ched4r, just outta curiosity, do you like write meth books or something??

.....that iz, "MATH" books....

i JUST DID IT BACKWARDS AND CAME UP WITH THE CORRECT ANSWER. gOOD PUZZLE.

WAY TOO MATHY 4 ME!

too hard for me.I'm not good at math.

I think there is some confusion over the counting of steps. If there were no steps, it would be flat ground. If there was one step, the ground has two levels, and there is no "middle step". If there are two steps, the ground has three levels, and the middle step is step 1 (counting the ground as step 0). From this we can see that there is only a middle step when the number of steps is ODD.

If you still don't believe this, try to picture 1 step, and think if there is a middle step:

_______________

____________|

Then, picture two steps:

______________

_____|

________|

And you can see quite clearly that there is a middle step in this case (you wouldn't call this 3 steps, would you? If you would, the first picture must therefore show 2 steps, and 1 step would be flat ground)

I agree with Canu, and get the answer to be 48 steps. It would only be 49 steps if one (erroneously) counted the ground as being a step.

If you still don't believe this, try to picture 1 step, and think if there is a middle step:

_______________

____________|

Then, picture two steps:

______________

_____|

________|

And you can see quite clearly that there is a middle step in this case (you wouldn't call this 3 steps, would you? If you would, the first picture must therefore show 2 steps, and 1 step would be flat ground)

I agree with Canu, and get the answer to be 48 steps. It would only be 49 steps if one (erroneously) counted the ground as being a step.

Sorry - text pictures got spoiled in the last post:

If you still don't believe this, try to picture 1 step, and think if there is a middle step:

_______________

___________ |

Then, picture two steps:

______________

_____|

________|

And you can see quite clearly that there is a middle step in this case (you wouldn't call this 3 steps, would you? If you would, the first picture must therefore show 2 steps, and 1 step would be flat ground)

If you still don't believe this, try to picture 1 step, and think if there is a middle step:

_______________

___________ |

Then, picture two steps:

______________

_____|

________|

And you can see quite clearly that there is a middle step in this case (you wouldn't call this 3 steps, would you? If you would, the first picture must therefore show 2 steps, and 1 step would be flat ground)

The ground is not a step. The ground is the ground. In order for there to be a middle step, there must by definition be an odd number of steps of at least 3.

Very good teaser. I didn't think to try to work backwards - that would have been a good hint.

Very good teaser. I didn't think to try to work backwards - that would have been a good hint.

way to much brain work needed!!!my brain hurts just thinking about it! but good one

i figured it out just like ched4r...i used x as the # of steps....great teaser..liked it....the whole monk thingy made it a lot more enjoyable...great job!

Every time there's a teaser like this there's a discussion about odd or even number of steps. Those who said if there is a middle step then by definition there must be an odd number of steps are correct. If you remember this you'll have an easier time getting the right answer the next time you run accross one of these even if you don't want to beleive it.

I didn't use any math, I just drew a picture as I was reading the teaser, and that made it really easy. Starting with a line that indicated the middle, there ended up being 28 steps above it, so that meant there had to be 28 steps below it, with the middle line being the last step I counted. If you do it that way, there really is no question of whether there could be a middle step or not, there just is and you count it and move on.

If there must be an odd number of steps to have a middle step, then which step is the middle one when there is only one step?

___________|""""""""""""""

If there are two steps, then the first step is clearly the middle step, since it is an equal distance from the top and the bottom:

_________------""""""""""""""

The confusion lies, I think, in the fact that, as you said Magicbankerm the ground isn't counted as a step, but it IS treated as a boundary, just as the top step is. Therefore, when finding the centre (and do be careful not to misuse the phrase "by definition") we have to use the half-way point between top and bottom, as you can see from the second diagram above. The first step is midway between the top and bottom, and is therefore "the middle" step.

___________|""""""""""""""

If there are two steps, then the first step is clearly the middle step, since it is an equal distance from the top and the bottom:

_________------""""""""""""""

The confusion lies, I think, in the fact that, as you said Magicbankerm the ground isn't counted as a step, but it IS treated as a boundary, just as the top step is. Therefore, when finding the centre (and do be careful not to misuse the phrase "by definition") we have to use the half-way point between top and bottom, as you can see from the second diagram above. The first step is midway between the top and bottom, and is therefore "the middle" step.

I don't understand why anybody felt this was hard. If you can count backwards, you can solve it with simple arithmathic. It was good that the teaser started us on the middle step, or it would have been even more easy.

robinmckenzie; you are misusing the words "middle step". What you are describing is the middle of the stairs, not the middle step. There is not a middle step when there are only one or two steps on a staircase. The idea is once you step up, you have taken one step. So the top is a step. The following picture (hope it works) is a representation showing that 2 is the middle step of 3 steps. Even though the top of step 2 is not directly in the middle of the stairs, it is the middle step.

------_________

----_|3

--_|2

_|1

------_________

----_|3

--_|2

_|1

Good one.

Okay I had to do this for myself...in my last post I put 28 and I meant to put 24 - it was bothering me so I signed back on to correct it... Now everything is right with my world and I can go back to work.

Have a great day everyone. And thanks again for the teaser - if I even did say thank you in the first place.

Have a great day everyone. And thanks again for the teaser - if I even did say thank you in the first place.

is he walking or climbing

From the midle step he walks up 24 steps-That's just the math. From there the answer depends as to what one defines as the middle step. I recall that median is defined as the middle value: If there are 'n' objects arranged in order the median is defined as the value at (n+1)/2 position if n is odd and the average of the two middle values if n is even. If n = 49 yields a median of 25 and if n = 48 yields a median value ((24+25)/2) of 24.5. Since there can't be a a physical step at position 24.5 n would be 49.

robinmckenzie u dont count the levels u coulnt the steps no one gives a **** if there are 3 levels there would have to be 3 steps for there to be a middle step so if there were two steps theres no middle step

p.s. how are we supposed to know that he is climbing 3 steps!!!!!!!!!! that is stupid how can we get it right if we have no idea that the fatso priest is taking 3 steps in 1 pace!?!?!??!??!??!?!

Jun 23, 2005

You know i am a bit confused.. not sure if 49 is correct or 48.

I went through the calculations as below. I sort of agree with 49 but cannot seen to go beyond 48.

Total = X steps

(x/2 + - 12 + 1 + (3* 9) = X

(x/2 + + 16 = x

((x+16) + 32)/2 = x

x+16 + 32 = 2x

x + 48 = 2x

x = 48

So u see, this was very simple, but I am ending at 48 while I do agree with 49. What do y'll think ?

I went through the calculations as below. I sort of agree with 49 but cannot seen to go beyond 48.

Total = X steps

(x/2 + - 12 + 1 + (3* 9) = X

(x/2 + + 16 = x

((x+16) + 32)/2 = x

x+16 + 32 = 2x

x + 48 = 2x

x = 48

So u see, this was very simple, but I am ending at 48 while I do agree with 49. What do y'll think ?

I think you are all nuts...

I must agree with Kukmeister. You all need to get a life and not ANALIZE things so much!

I got 48 steps just like nako, but then I have to agree that if there is a middle step then it must be odd numbered. I worked forward. When it says he covered 27 steps (9 x 3) and someone said 'Simple, it's 28 steps he covered' I don't get that. How does 27 steps equal 28 steps?

nako, you're on the right track, but to find the center (median) your formula is (x+1)/2 . . . not (x/2). That's probably why you ended up with an even number.

To have a middle, you have to have an equal number of items on either side of that middle. Therefore, you cannot have an even number when you are calculating with/from a middle point.

Great teaser!

To have a middle, you have to have an equal number of items on either side of that middle. Therefore, you cannot have an even number when you are calculating with/from a middle point.

Great teaser!

You guys are spending way too much time on this one!

Too difficult for me....

This is simple algebra:

Number of steps up from middle

=======================

+27 (3*9) steps up

+1 step up

-12 steps down

+8 steps up)

Don't forget the middle step

+1 the middle step

x=2(3*9+1-12++1

x=2(27+1-12++1

x=2(24)+1

x=49

Number of steps up from middle

=======================

+27 (3*9) steps up

+1 step up

-12 steps down

+8 steps up)

Don't forget the middle step

+1 the middle step

x=2(3*9+1-12++1

x=2(27+1-12++1

x=2(24)+1

x=49

Hee, my equation got mangled by emoticons...

x=2(3*9+1-12+8 )+1

x=2(27+1-12+8 )+1

x=2(24)+1

x=49

x=2(3*9+1-12+8 )+1

x=2(27+1-12+8 )+1

x=2(24)+1

x=49

I have the most brilliant equation in the world.... Well, it used to be, at least. I think it still is. But I have the most brilliant equation!

= 2

= 2

AND ANOTHER!

+ =

+ =

i agree with dudeman81

great teaser though, just simple math. i worked backwards and got it right away

great teaser though, just simple math. i worked backwards and got it right away

Jun 23, 2005

I liked it. It was simple if you worked it backwards.

I missed it, but saw mistake when i worked it out looking at answer, I loved it however was real good.

Have you posted this somewhere else too? I've done this one before....but it wasn't here.

There are only 5 steps, not 48, nor 49, just 5.

There are only 5 steps to HIS ROUTINE OF GOING UP!

Step 1 - he climbs up and meditates,

Step 2 - up 8 til the bird sings,

Step 3 - walks down and gets pebble,

Step 4 - up 1 and tosses pebble,

&

Step 5 - takes remainder 3 at a time.

See, five steps. Right???

There are only 5 steps to HIS ROUTINE OF GOING UP!

Step 1 - he climbs up and meditates,

Step 2 - up 8 til the bird sings,

Step 3 - walks down and gets pebble,

Step 4 - up 1 and tosses pebble,

&

Step 5 - takes remainder 3 at a time.

See, five steps. Right???

teri that was AAWWEESSOOMMEE!!!!!!!!!!!!!!!

BBOORRIINNGG! But I figured it out! Great teaser!

You guys debating 48 steps are nuts! The ground is the ground and a step is a step. Silly people.

sorry- i agree with the people who say 48 steps

I think too much thought is being put into this.

Just have fun with these!

Just have fun with these!

i had a lot of fun figuring it out!

I'm glad you liked it.

I totally missed the "middle" step. I got 48 too.....

Confusing!!!!

teri, it would only be that if it was in trick category, lol

I got 48 steps: close enough but I'm still not sure where that last step comes in. Good teaser anyhow!

You have to count in the middle step.

49, very tricky but I got it.

Great teaser.

49, very tricky but I got it.

Great teaser.

Where's the teaser? As I read the story I'm counting M+8, M-4, M-3, M+24. So 24 above the middle and 24 below plus the middle for 49. There should be at least some math--this should be in the logic category.

At any rate, too easy to be a teaser.

At any rate, too easy to be a teaser.

Since there is a middle step, we must have an odd number of steps. The middle step would be (S+1)/2 where S = total number of steps, so:

(1/2 [S+1] + 8 - 12 + 1) + 9*3 = S

1/2 S - 2.5 + 27 = S

1/2 S + 24.5 = S

24.5 = 1/2 S

S = 49 steps

(1/2 [S+1] + 8 - 12 + 1) + 9*3 = S

1/2 S - 2.5 + 27 = S

1/2 S + 24.5 = S

24.5 = 1/2 S

S = 49 steps

OK - draw 2 steps: ground, step, top landing ...

_______

_______|_x_

___________|_______

If you told me to stand on the middle step, I know where I would stand! There i am (I hope) at the 'x'. To my thinking, 'middle step' requires an EVEN number steps, so the answer must be 48 steps!

_______

_______|_x_

___________|_______

If you told me to stand on the middle step, I know where I would stand! There i am (I hope) at the 'x'. To my thinking, 'middle step' requires an EVEN number steps, so the answer must be 48 steps!

I think this illustrates the difference between maths and logic. Palladin correctly uses maths to solve an equation.

However, if logic is used, we cannot assume, as Javaguru (I think) incorrectly does, that there is an odd number. Logic must be built on correct assumptions. From my diagram above, we see that 'middle step' is a concept which, by normal use of language, requires an EVEN number of steps.

Maybe Javaguru is correct, then, that this is more logic than Maths!

However, if logic is used, we cannot assume, as Javaguru (I think) incorrectly does, that there is an odd number. Logic must be built on correct assumptions. From my diagram above, we see that 'middle step' is a concept which, by normal use of language, requires an EVEN number of steps.

Maybe Javaguru is correct, then, that this is more logic than Maths!

First of all, the years-long discussion of 48 vs. 49 steps saddens me; the middle step must be a whole physical step added to the equal number up and down, not fractional, ergo 24+24+1=49. I admire all the people who either worked it backwards or by way of an algebraic formula; I seem to be the only one so far who's gotten the answer by trial and error (picking a # of steps and plugging in the monk's ritual!) As for the explanation, I have no idea where you got step 33 as a starting point!!! What did I miss? How were we supposed to know that?? Will someone please explain? Thank you.

Wading into the odd/even steps debate...

I mathematically calculated from the middle step that there were 24 above the starting point, therefore double that gives 48.

The difficulty is that an even number gives a middle number of treads with two ground levels, and an odd number gives a middle number of step faces.

______

5/__

4/__

3/__

2/__

1/________

______

4/__

3/__

2/__

1/________

My interpretation is that the even number (4 steps above) gives a step that is an even height from the upper ground and the lower ground. Therefore step 2 is in the middle, and the even number of steps theory is correct.

If you look at the case of the 5 step stairs above according to the 49 step proponents, if you want to climb up to the middle step you take 3 steps. Then you have a further 2 to get to the top. So that doesn't work. Let your fingers do the walking on the diagram above. "Walk" along the "ground" then take 3 steps up and then place your other finger(foot) on the third step as well. Then continue up the stairs, and you only take 2 more until you reach the top.

I guess you need to ask if you want to start in the middle, or if you want to start with the same number of steps above and below.

I mathematically calculated from the middle step that there were 24 above the starting point, therefore double that gives 48.

The difficulty is that an even number gives a middle number of treads with two ground levels, and an odd number gives a middle number of step faces.

______

5/__

4/__

3/__

2/__

1/________

______

4/__

3/__

2/__

1/________

My interpretation is that the even number (4 steps above) gives a step that is an even height from the upper ground and the lower ground. Therefore step 2 is in the middle, and the even number of steps theory is correct.

If you look at the case of the 5 step stairs above according to the 49 step proponents, if you want to climb up to the middle step you take 3 steps. Then you have a further 2 to get to the top. So that doesn't work. Let your fingers do the walking on the diagram above. "Walk" along the "ground" then take 3 steps up and then place your other finger(foot) on the third step as well. Then continue up the stairs, and you only take 2 more until you reach the top.

I guess you need to ask if you want to start in the middle, or if you want to start with the same number of steps above and below.

5 and 4 step staircases redrawn

______

......5/__

.........4/__

............3/__

...............2/__

..................1/________

______

......4/__

.........3/__

............2/__

...............1/________

______

......5/__

.........4/__

............3/__

...............2/__

..................1/________

______

......4/__

.........3/__

............2/__

...............1/________

This was a good puzzle, and a heck of a lot of thought put into it!

Gee, for that matter, a heck of a lot of thought put into the explanation comments!

For those of you who solved it, bravo!

Gee, for that matter, a heck of a lot of thought put into the explanation comments!

For those of you who solved it, bravo!

49 steps and 6 years running, this one is wornout.

Needless to say, since I will admit I'm math challenged, and not particularly logical, I didn't get this right. I had more fun reading all the comments over the years. Thanks everybody for the fun reading.

Easy but fun. I worked from the middle step and got it in less than a minute. So much controversy over a simple problem. Steps are steps, ground is ground....not a step.

When it cums 2 math, I am terrible!!!!!!!!!! Interestin teases!!

I got 48, but I can't see what I did wrong though.

How do you know that there are precisely 49 steps?

The conditions state that the last 9 paces are taken three steps at a time. However, the last pace may be one, two, or three steps, yet it must be taken in order to reach the top. Since you can't be sure of the number of steps in the last pace, how can you determine the total?

The conditions state that the last 9 paces are taken three steps at a time. However, the last pace may be one, two, or three steps, yet it must be taken in order to reach the top. Since you can't be sure of the number of steps in the last pace, how can you determine the total?

Aug 02, 2011

So you mean like 25 is the middle step and it's not counted when you count the step upwards and downwards?

I don't do math teasers so I don't care how many steps there are!

After 9 years, I think it is time to retire this teaser. I understand some people have not seen the older teasers and they are nice to have appear on occasion, but looking at the dates this has been Teaser of the Day on several occasions.

Why not let some of the newer teasers (not the brand new ones that sow up on the right of the screen) have a go at being T.O.T.D.?

IN the 3 years since this was last featured, there must have been at least 1 or 2 decent teases submitted, if not, what are we all doing here?

The teaser itself was great, challenging enough to not be obvious but not so hard it was impossible to solve.

Why not let some of the newer teasers (not the brand new ones that sow up on the right of the screen) have a go at being T.O.T.D.?

IN the 3 years since this was last featured, there must have been at least 1 or 2 decent teases submitted, if not, what are we all doing here?

The teaser itself was great, challenging enough to not be obvious but not so hard it was impossible to solve.

I'm with Babe on this one.

Bravo to HABS for telling it like it is.

I'm with Babe on this one - math teasers are far from being my favorites, but I also am kind of ashamed of myself to admit that! But thank you, HABS for your thoughtful comment - you always come through with something substantial like that! Hi Doehead and nice that you complimented HABS. Loved all the comments today!

whoops, I think I goofed on my last comment! Sorry!!

Nobody will catch your goof. Still love math teasers, and they are always new to an old brain like mine.

, never mind, it's old news . . .

YESS! I got it! I liked this one a lot.

nicely done

Once you dope out that, in order for there to be a middle step, the total number of steps must be odd, and that in any series that ends in an odd number, the midpoint of the series must be (n + 1)/2, where "n" is the last number in the series the algebraic solution to the problem becomes obvious. It took me a while, though, to reason this out.

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