How Many Triangles Are There? Learn The Formula For Any Size!
Вставка
- Опубліковано 16 гру 2024
- How many triangles are formed in a grid of equilateral triangles with N triangles in its base? The video shows a pattern in the case of n=4 and presents a formula for the general case.
Blog post (text explanation): wp.me/p6aMk-4sO
There are many ways to prove the formula, links in this stackexchange discussion: math.stackexcha...
Ask Dr. Math explanation: mathforum.org/l...
Sum of numbers 1 to N visually: • Beautiful Pattern In T...
Sum of numbers 1 to N algebraically: • Gauss discovered this ...
If you like my videos, you can support me at Patreon:
/ mindyourdecisions
Connect on social media. I update each site when I have a new video or blog post, so you can follow me on whichever method is most convenient for you.
My Blog: mindyourdecisi...
Twitter: / preshtalwalkar
Facebook: / 168446714965
Google+: plus.google.co...
Pinterest: / preshtalwalkar
Tumblr: / preshtalwalkar
Instagram: / preshtalwalkar
Patreon: / mindyourdecisions
Newsletter (sent only for big news, like a new book release): eepurl.com/KvS0r
If you buy from the links below I may receive a commission for sales. This has no effect on the price for you.
My Books
"The Joy of Game Theory" shows how you can use math to out-think your competition. (rated 3.9/5 stars on 32 reviews)
amzn.to/1uQvA20
"The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias" is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 4.6/5 stars on 3 reviews)
amzn.to/1o3FaAg
"Math Puzzles Volume 1" features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 13 reviews.
amzn.to/1GhUUSH
"Math Puzzles Volume 2" is a sequel book with more great problems. (rated 4.3/5 stars on 4 reviews)
amzn.to/1NKbyCs
"Math Puzzles Volume 3" is the third in the series. (rated 3.8/5 stars on 5 reviews)
amzn.to/1NKbGlp
"40 Paradoxes in Logic, Probability, and Game Theory" contains thought-provoking and counter-intuitive results. (rated 4.3/5 stars on 12 reviews)
amzn.to/1LOCI4U
"The Best Mental Math Tricks" teaches how you can look like a math genius by solving problems in your head (rated 4.7/5 stars on 4 reviews)
amzn.to/18maAdo
"Multiply Numbers By Drawing Lines" This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 5/5 stars on 3 reviews)
amzn.to/XRm7M4
I'm fairly confident there are more than 4 triangles.
+John Morrison Not 100% sure though, but it's gotta be more than 2.
+John M 2? Hmh... Seemed to me there was about one 1/2 triangle... Are you sure you aren't just trying to show off by saying a number off the grid?!
John M knfyhbf/Cuyfchydfju NBC's
nah, its more than 0 triangles
how’d u figure that out? I thought there were less than -6 triangles
I see 27
+xisumavoid you watch this? nice ;-)
Hi X
+xisumavoid Nice X
+xisumavoid They totally got me with the upside down triangle in the middle of size two. I missed that one and got 26 :(
Ooh
I would say...10+6+6+1+3+1=27.
10 small ones tip up, 6 small ones tip down, 6 "double" ones tip up, 1 double one tip down, 3 triple ones tip up and the big one.
The question is similar to the square question, but here you don't add square numbers but tringle numbers and that twice: 1+3+6+10 for tip up and 1+6 for tip down.
In such case, no of triangles from each three side will be equal.
Then let n be the no of triangle from each side:
Then by using follwing table, you can easily find the no of triangles.
n No of triangles
1 1
2 5
3 13
4 27
5 48
6 79
7 118
8 170
So, in this problem n= 4, thus no of triangles= 27
Thanks
16 small triangles (1 unit ea.)
7 medium triangles (4 units ea. - 1 upside down)
3 large triangles (9 units ea.)
1 full triangle (all units)
27 total.
You are a hero
@@arianapowell8370 Well thank you... but I was just thinking if we're really looking at a 3D pyramid with 4 identical sides, then my estimate of 27 triangles is wrong! Not to mention the problems with 4D or more!!!
I liked the graphics and explanation, and agree with the final result. But found a certain "leap of faith" at the end to generalize the case n- even to n- odd in terms of the "floor function". Using a different method, my results agree, but for the case where n is odd, I find the answer in a different form, to be (n+1)(2n^2+3n-1)/8.
Your answer is correct for Odd nr (1,3,5,…,2k+1).
The video gives the answer for the pair nr (2,4,6,8,…,2k)
We subtract 1 from numerator of pair nr formula for odd nr.
The unified formula is [n(n+2)(2n+1) - Mod(n,2)]/8
Size 1: 10 up,6 down total 16
Size 2: 6 up, 1 down total 16+7=23
Size 3*: 3 up total 16+7+3=26
Size 4*: 1 up total 16+7+3+1=27
*No downs
Welcome to the channel where counting is taught in a tough way
Lol
Solving this puzzle I've found some interesting things that I have never seen before. Can't put the formula here, but the pattern remains:
For polynomials of degree 0: f(x) = f(x-1)
For polynomials of degree 1: f(x) = 2*f(x-1) - f(x-2)
For polynomials of degree 2: f(x) = 3*f(x-1) - 3*f(x-2) + f(x-3)
For polynomials of degree 3: f(x) = 4*f(x-1) - 6*f(x-2) + 4*f(x-3) - f(x-4)
For polynomials of degree 4: f(x) = 5*f(x-1) - 10*f(x-2) + 10*f(x-3) - 5*f(x-4) + f(x-5)
etc
Where the coefficients follow the pascal's triangle.
Have you ever seen this before? Where can I find more about this? And I want to know if this is useful for something also.
Genious! Talk about this at the University. Maybe nobody hasn't found it yet. Beware! This idea is YOURS. Don't let them steal your idea! 👏🏻👏🏻👏🏻
To find the number of triangles that only point upwards is a formula that took me a day and a half to calculate. It uses sigmas within sigmas
so a while ago i formulated this and got something much more complex.
(Sigma(lim=R, X=1) of X(R-X))+(Sigma lim=R, x=4 of x-3)+R^2
where R=how many rows there are
This video is really helpful after watching this I tried solving so many other and got the right answer
armyyyy!!💜💜
Thank you for the simple solution!
I solved it by making a recursive form
a_n is the number of triangles for a big n sided triangle
a_n = 3a_n-1 - 3a_n-2 + a_n-3
Then make the general formula using the charestersitic roots formula
We learnt this in math olympiad it is called triangle numbers alternatively you could use gauls theorem but that is slow
I like this. Getting the answer was easy. The formulas are what I liked. Sadly I have forgotten this. I need to study. I believe he should have stayed to round to the nearest whole number instead of saying to round down though.
1:59 It's 27 total for those who don't want to waste time looking for this exact problem in your tests.
If wanting to skip the rounding, and doing it by direct substitution it can also be wrapped in the formula f(n)= (2*n*(n+1)^2+n^2-(1/2)+(1/2)*(-1)^n)/8... Just saying...
There is ready made formula
n(n+2) (2n+1) /8
Where n =Number of Horizontal Line
In this first case n is 4
for 5 number of rows. = N so formula
5 square + 4 square +3 square + 2 square + 1 square = 55
then reduce from 55
(n-3) square + (n - 2)
So
(5-3) square + (5-2)
= 4 + 3
=7
Hence number triangles = 48
Quick and easy
Singles = 16
Next size up = 7
Next size up = 3
Next size up = 1
27 triangles
Singles = 16
Next size up = 7
Next size up = 3
Next size up = 1
27 triangles
27🙌
Seen many of these puzzles
But am here to get to know the formula :-)😇
ببت
S1: 1 + 3 + 5 + 7 = 16
S4: 1 + 2 + 3 + 1 (upside-down) = 7
S9: 3
S16: 1
Total: 16 + 7 + 3 + 1 = 27.
for the square is n (n+1)(2n+1)/6
I think there is a mistake, because if you take an odd number, for example three, you are going to get: T = 3.(3+2).(2.3+1)/8 = 3.5.7/8 ... an irreducible fraction. This method just is helpful when ¨n¨ is an even number.
You forget that he mentioned you have to round down. So a triangle with 3 triangles at the base would have 13 total triangles within it
yup. You are right! how can 105/8=13 when its 13.125.....
i guess we have to take approximates.... as 13.125~~13
MattAndRob HumorShow And what happens if ¨n¨ is a big number? Tanishq reports an error of 0.125 .The question is: Would this error increase and be more big than one? My english is basic, be nice.
+Albanovaphi7 I don't think so. The error is always going to be 0.125
+Albanovaphi7 He uses floor function, it just takes the integer part
nice video sir g like it
lol verified.
n=number of upside up triangles in the bottom row n(n+.5)(n+2)/4
27 im pretty sure
17
Correct answer: none, my eyes are closed.
You're right. :D
The thumbnail asks how many do you see, not how many there are, so he is correct.
ua-cam.com/video/98gSDCZrcdw/v-deo.html
I tried ... ,sequence sum, in this way!
How did you write the comment then?
If n is odd then number of triangle is (n(n+2)(2n+1) -1)/8
This is nice.
I counted them to be 26 and checked the correct answer. Later closely checked for the downward inner size 2 triangle to get the correct answer as 27.
1+4+9+16
Thanks!! I counted 26 and didn't know where I went wrong.
Did you miss the upside down triangle .. made up of 4 small triangles
With a bit of lengthy calculation, I got the following outcome:
If n is the number of equal divisions into which each side of the outer triangle is divided, then
(n/8)(2n2+5n+2) if n is even
Total number of triangles in such a figure =
{(n+1)/8}(2n2+3n-1) if n is odd
In the above case, n = 4. Hence total number of triangles in the figure = (4/8)(32 + 20 + 2) = 27.
n=5 / 5×7×11 is nt divisible by 8
Rounding down. The reason you round down is that downfacing triangels are much fewer than upfacing.
So the answer is 240.
5x7x11=385
385:8=45,625
@@Jo_Es_Chess_Channel 385 : 8 is not equal to 45,625 , but to 48,125
[х] means the whole part of x, so [5*7*11/8] is equal to 48
Luca Piersante whatever it's still decimal 😂
(n*(n+2)*(2n-1))/8).
how did you derive this formula.
n=? Please explain
n=number
n (n+2)(2n+1)/8 if even
n (n^2+5n+3)/8 if odd
“suhailath kp”, why?
ua-cam.com/video/7PzdKuwU1EM/v-deo.html
that was so cool i did not know there were that many triangles :) cool!!!
This is the exact formula without rounding:
f(n) = (2n(n+1)(n+2) - n(n+1) - n)/8 + ((-1)^n - 1)/16
Or:
f(n) = n(n + 2)(2n + 1)/8 + ((-1)^n - 1)/16
It gives: 1, 5, 13, 27, 48, 78, 118, 170, 235, 315, 411, 525, 658, 812, 988, 1188, ...
guardofrr
Sloane's A002717.
Jep.
at first i was like: 16 triangles, then i saw 11 other triangles and that added up to 27
Excuse me sir, you are right. Of course this formula is for even N. I see Mike Andreoli gives the formula for odd numbers.
My senior HS math teacher tried to teach me this and years later I have looked it up ....answer at time 4:08
the fornula [n(n+2)(2n+1)/8] is just for any even number n, the formula for odd number n is [(2n^2+3n-1)(n+1)/8]. you can check for this.
Yes, but the formula used by Presh is actually
⌊n(n+2)(2n+1)/8⌋ ---> ⌊ ... ⌋ is the floor function
which works for all values of n.
This works because [n(n+2)(2n+1)/8] - [(2n^2+3n-1)(n+1)/8] = 1/8 = 0.125.
This means that if you plug an odd number into first formula, you get a value that is 0.125 more than correct value. When n = 3, n(n+2)(2n+1)/8 = 27.125. If you apply the floor function to this you get 27. This will work for all odd values of n. Alternatively, you could use the ceiling function on the second formula:
⌈(2n^2+3n-1)(n+1)/8⌉
I'm fairly confident its 27
Illuminati confirmed.
may the triangle rule
😂😂
What if n is odd, will the formula change? Will 8 remain constant no matter what is the value of n? Take 5 as n for example, how will you solve that with this formula?
There is a slight change in the formula. We subtract 1 before dividing by 8. [n(n+2)(2n+1)-1]/8.
I start with the big and go smaller and smaller, so i just write the numbers i see(both up and down will be added in, i just size them): 1+3+7+16=27
Our maths sir gave us 2 HWs of finding how many triangles r there...the student with the correct answer will get a prize. And I saw this vid in his history :D So,came accross this vid and finding it helpful TYSM!!! Can't wait to win a prize
how will you solve if n is an odd?
If n=3 then, it would be like-
3(5)(7)/8=105/8 so how can be the numbr of triangles would be in decimal....?????
Take -1 with numerator
The symbols around the equation tell you to round down like he says in the video.
is it valid to multiply together the number of lines inside the triangle? this one had 3 slanting one way, times 3 slanting the other way, times 3 horizontal, which gets you 27. haven't checked if this works with other sizes or if it just happened to add up for this one but..it did work..and i counted them all to make sure too
That won't work. First, in this triangle there are 4 lines slanting one way, 4 slanting the other way, and 4 horizontal. So you might be tempted to think that number of triangles = (n-1)^3 instead. But this won't work, because when n = 5, number of triangles = 48 which is not a cube.
Second, not all combinations of such lines form a triangle. If you take 3 lines (one slanting one way, the other slanting the other way, and a horizontal one) that all intersect at the same point, then you cannot form a triangle. Another problem is if 2 of the chosen lines don't intersect anywhere on the triangle, but somewhere off the triangle.
@@MarieAnne. I get what you're saying but I did specify using the number of lines INSIDE the triangle, i.e. excluding the ones that make up the perimeter of the largest triangle because those are already implied to make one triangle. Does that change anything?
@@jaimecarter3988 But other than the largest triangle, many of the other triangles ARE actually formed by one or two of the perimeter edges. Others are formed by 3 inside edges. And there are cases where some combination of these lines (including 3 inside edges) will not form triangles. I'll include a link showing examples in the post below if I can.
@@jaimecarter3988 Oh well, forget the link. No matter what I try, it keeps getting deleted. I hope you understand what I mean from my description.
Sir i have a problem with this formula when it works for n=odd then there is some error ... and this error repeats itself for n=odd...
I explain it as if
When n=1
=1×3×3/8=1.125
When n=3
=3×5×7/8=13.125
When n=5
=5×7×11/8=48.125
When n=7
=7×9*15/8=118.125
And so on the error of .125 goes on its my sugestion that can we afford to neglect these .125 value so will it not be better if the formula for n=odd would rather be
n=odd
Total triangles be..
=(n(n+2)(2n+1)/8 -0.125)
It would rather be accurate even..
I will be highly obliged if u plz make these correction..
And a very very congratulations for the innovative formulas u make
Thank you..
f(n) = (2n(n+1)(n+2) - n(n+1) - n)/8 + ((-1)^n - 1)/16
In the sub titles his name was translated or depicted as pressure locker😂😂
it is wrong for odd no of sides.for 3 sides ans in decimal. as per formula....3*5*7/8=13.125.what u say for this decimal????.
The formula has a step value function which means the closest integer less than or equal to the answer you get. Thus the step value of 13.125, which you got, is 13, which is the number of triangles.
The formula is wrong it only works for even numbers, go ahead try it for 1, 3 ,5 etc.
I was able to create a new recursive formula though where you can calculate the amount of triangles of the next triangle (where the base is increased by 1) with the amount of of triangles of the previous 4 big triangles. This one worked flawlessly.
No, formula works for odd numbers because of the floor function.
When n is even, # triangles = n(n+2)(2n+1)/8
When n is odd, # triangles = (2n^2+3n-1)(n+1)/8
But guess what? n(n+2)(2n+1)/8 - (2n^2+3n-1)(n+1)/8 = 1/8 = 0.125
So what happens when you plug an odd number into the even number formula? You get a value that is 0.125 more than the actual number of triangles (every time), which means that when you take the floor function (as Presh mentions in video), you get correct answer whether n is even or odd.
# triangles = Floor(n(n+2)(2n+1)/8) = ⌊n(n+2)(2n+1)/8⌋
There is no one correct answer to the question "How many triangles do you see?" If I close my eyes, the answer is zero. If I check every possible triangle there could be seen, the answer is 27.
ft55555 wow
ft55555 wow
ft55555 wow
Sir , isn't there any formula for this type of qns?
Doesn't anyone think the question to this puzzle is not specific enough? it was explained the different sizes of triangles pointing up and down, but what if we also have to think of the triangles pointing left and right? Wouldn't there be more triangles to count? so the question should ask how many triangles pointing up and down, because equilateral triangles point in 3 different directions then each triangle count as 3.
Although it might be facing left and right, if you were to outline the triangle, and it overlapped, it’s still the same triangle. The direction is relative to how you look at it. The outline is what matters
I am surprised. I figured this out a day ago and today I see this video. Coincidences
Its explained in my good way.
I was off by 3. Need more of theses to practice
Adam Jensen took one look at this and flatly said "I never asked for this."
Is there a formula for a similar problem with a square?
If it's like a grid of squares, then formula is (n²+n)(2n+1)/6
If we put n=3 or n=5 then your formula does not apply. I mean then the answers are not 13 or 48
PLEASE CHECK
The formula has a step value function which means the closest integer less than or equal to the answer you get. If you put n = 3, you get 13.125. Thus the step value of 13.125 is 13, which is the number of triangles. The same applies for all odd n.
Technically, that formula only works above 1. Because (n(n+2)(2n+1))/8 where n=1 comes to 9/8, which =/= 1
Technically, the formula used by Presh does work.
He is NOT using n(n+2)(2n+1)/8. Instead he is using ⌊n(n+2)(2n+1)/8⌋
where ⌊...⌋ indicates the floor function.
The formula n(n+2)(2n+1)/8 works only for even values of n. However, when we use an odd number (such as n = 1), we get a value that is 1/8 (or 0.125) higher than correct value. (n = 1 gives 1.125, n = 3 gives 13.125). But when you use floor function, you do get correct result.
Floor(9/8) = Floor(1.125) = 1
Floor(13.125) = 13
@@MarieAnne. I did not know that, thank you for clarifying. I've only gone up to high school calculus, so never learned about floor functions.
@@MarieAnne. formula should have been calculated for odd and even number. The difference in these formulas numerator is exactly one. So rounding may work but it not the right formula. Subtract MOD(n,2) from numerator gives the unified formula.
Ez, and classic challenge, it's 27
16 small ones (1x1 obvious)
4 medium ones (2x2 if u fold it along these, u get a Pyramid)
3 medium ones (2x2 placed at the center of each edge)
3 large ones (3x3)
1 big one (4x4, the intire thing)
+Cube does Stuff 7 medium 2x2 yes i concur.
There’s a great Star Wars reference in his videos
Presh ‘Skywalker’
Depends on how you look at your possibility to see those triangles in the picture before you look at the video. There Is a big chance you will not see the same thing as this guy explained in what he saw.
This formulae is not the same for odd and even base. Its different for odd base and for even base..For odd base deduct 1 in the numerator before dividing with 8.
YOU FORGOT THE ONES AT AN ANGLE FROM UP OR DOWN AT LEAST THREE EASY ONES I SEE BUT I HAVE BETTER THINGS THAN THIS TO DO THAN LOOK AT ALL OF THEM PLUS TILT RIGHT OR LEFT
Perhaps the simple way to solve such counting problems, is to proceed systematically. Count as many triangles as you can of side length 1, then count as many triangles as you can of side length 2, then count as many triangles as you can of side length 3, and finally of side length 4. There is tremendous value, benefits, and gains to be had in adopting a systematic, thoro approach, rather than an ad-hoc approach, to doing anything and everything in life. It always far out-performs the winging approach.
that problem has been generalized by induction.
but could it be solved by deduction?
Khub Sundor Video
The formula won't work when 'n' is odd number like 3. It needs to be modified by subtracting 1 in numerator when n is odd
Of course it won’t work when it’s odd, you would never have a triangle with odd smaller triangles anyway… look at it for a second gosh
thanx. needed this for a long time. but explanation should be little more illustrative.
How r u generalizing the triangle counts facing down sir? Can someone pls explain?
Can a formula be derived for the number of triangles in say nth level of Sierpinski triangle?
I might make a video on deriving the number of triangles for sierpinski sometime. i've derived the step-by-step formula for this basic triangle counting. ua-cam.com/video/7PzdKuwU1EM/v-deo.html
Let me just pretend that I understand the later part of the video
Here is something you'll understand Abhisek Pal. An easy step-by-step derivation of the formula. ua-cam.com/video/7PzdKuwU1EM/v-deo.html
Thanks u made me learn a lot from that vidieo
What is n in the formula ?
Im not sure, but it feels like i could get in trouble for saying it...
Thank you soo much
Its really nice
please make a video on counting in 3d (same fig but in 3d ie prism over prism, then how many triangles
Could you show how to come up with this formula? The Up triangles are pretty easy but I have trouble with the down ones :/
+Patrick Wienhöft
We know the numbers are: 1,5,13,27,48,78...
So we are adding these numbers: 4,8,14,21,30
And to get those numbers we are adding these: 4,6,7,9
And to get those numbers we are adding: 2,1,2...
The last series of number is 2,1,2,1... There is a pattern there. There are 3 series until I get to a "logic pattern", so the formula is something like this: AX^3+BX^2+Cx. We are adding 2's and 1's so we will have to round the odd numbers. We can do an equation system:
8A+4B+2C=5
4^3A+4^2B+4=27
6^3+6^2+6=78
A = 1/4
B= 5/8
C= 1/4
So the formula is: 1/4x^2+5/8x^2+1/4x = x(x+2)(2x+1)/8
Sorry for my bad English.
+Train-Tech-Education Yep, it's actually the formula he shows at the end of the video.
+Train-Tech-Education It's the same formula, but it's not factorized.
EXPLAIN why that formula and why rounding down!
10 size 1 triangels right side up
6 size 1 triangels botms up
6 size 2 triangels right side up
1 size 2 triangels botms up
3 size 3 triangels right side up
0 size 3 triangels botms up
1 size 4 triangel right side up
0 size 4triangls botms up
= 27 triangels
I wasn't that far off. I guessed 22.
Mainly because I forgot about the three sided ones and the second two on the bottom row facing up.
If you make a slight adjustment, you can get the right number for all n without rounding down.
Total = [n (n + 2) (2n + 1) - 0.5 ((-1)^(n + 1) + 1)]/8
- 0.5 ((-1)^(n + 1) + 1)/8 subtracts 1/8 every other turn.
At 0:10 I saw 27 (btw I’ve done this riddle before)
Δ
I see 28.
i too
i too
T24 #meetoo
28 STAB WOUNDS
I don't know if I'm the first one to have """discovered""" this formula, but I found a formula that works for this problem that doesn't require any floors or anything like that, just pure algebra:
(4x^3 + 10x^2 + 4x - 1 + (-1)^x)/16, where x is the size of the largest triangle.
It's basically two formulas frankenstein'd together, one being the sequence of tetrahedral numbers:
(x(x + 1)(x + 2))/6,
and A173196 from OEIS:
(4x^3 + 6x^2 - 4x - 3 + 3(-1)^x)/48
Not sure how helpful this actually is, but who knows?
tilt the triangle, you can find more (moving the base clockwise, twice) then repeat your method, you should come up with more triangles…
Wouldn't they be the same triangles already identified but rotated/from a different angles though?
If you count the size 2 triangle facing down then I suggest an argument that includes all the triangles rotated 120 degrees are also counted. Applying some lateral thinking 😂. Answer 81
Amazing talent in explaining the concept.
Well in the thumbnail were 29
No, there were 30. The 27 in the picture plus the 2 triangles in the letters "A" and the word triangle
@@cristiluchian3567 oh hey i dont remember writing this comment. anyways thats like saying the word "sentence" is a sentence which doesnt really make any sense. like you wouldnt stare at the word "triangle" and go "ah yes that is a triangle". but i guess you have a point kinda
Thanks for the video!!
I did figure it out, yes. Is it right? Who knows?
Zero cause there all illuminaties
did u pulled and i^2 =n (n+1)(2n+1)/6
I have another formula for calculating this and mine is so easy to calculate
What's the formula??
2× small triangles - 5
2×16-5=32-5=27
that doesn't work for the other sized triangles
Whats does "rounded down" mean ?
When you get a number with decimals, but want to get rid of them you can use either round up and round down. If you have 4, 3, 2, 1 or 0, you have to round down, and if you have 5, 5, 7, 8, or 9 you have to round up.
Example; 10.4 ~~ 10
Example; 10.5 ~~ 11
Hope this helped. ^-^
Snoopy Knutsen Thank you very much ! :)
+Snoopy Knutsen Not quite. "Rounded" would mean to round up when you have 0.5 or more and round down when you have less. But "Rounded down" would mean to round down no mater what, e.g. 10.8 would go to just 10, not 11, 10.
I see 27
16 are small
7 are medium
3 are large
1 is GIANT!
Total fomula is correct in case n is even number, in case odd number isn`t.
I'm 68years old. When I get to around 20 my eyes start crossing then I can't find my way back. I did see 3 big squares, does that count for anything?