Awwwww! Thank you soooo much Mr. Puzzle. I am so excited you picked my comment, really honored. My students next school term are going to go nuts when I show them the video and they get to try the puzzle for real! ❤️ 😃

"Knowing the solution cannot be undone!" I think you've greatly underestimated my ability to forget things with precision and *speed*

If the largest circle has 13 times the area of the smallest circle, then the radius of the largest is sqrt(13) times larger. So if the side of the square is 18.59 times the radius of the small circle, it is 18.59/(2*sqrt(13)) = 2.578 times the diameter of the largest.

This makes me wonder if squared squares would make hard puzzles. Would be something aesthetically and mathematically pleasing for sure.

My first strategy for solving this: "Hmmm, maybe we can flip the pieces?" Yeah, I'm not the sharpest tool in the shed...

13:24 The official sound effect of victory.

I take advantage of the fact that there is nothing to win this week to send you my sincere congratulations for all your videos! Keep on puzzling!

The question that comes to my mind: Is this the only solution? Or, is it by definition, only solvable one way?

Solution to your small exercise: The side length of the frame is 2,577969.. ~ 2,58 times the diameter of the biggest (13) circle.

Congrats to this week's giveaway winner! Very well deserved. What a heartwarming comment!

I thought the ultimate packing puzzle was fridges and freezers after a grocery shop. And then there's this. There is definitely an algorithm to solve this faster...but I have no time for figuring this out...

I figured you would have to put smaller pieces in the corners to minimize space lost on the 90 degree angle

It would be interesting to see Mr. Puzzle return to puzzles like this after a few months to see if he can do them faster or if it's about the same.

I work in upholstery and one part of my job is to try and fit as much pieces I have to cut into a set amount of fabric material, I guess you could say I do packing puzzles all day at work sometimes 😂 yesterday I didn’t think what I have to cut would fit out of the piece of material I had but after trying 4 times it actually all fit inside the material

My thought would have been to start with smallest circles in the corners

Oh! I would think the 13 will go in the middle, because not even the one fit's in that corner space. The straight borders likely always do maximum damage to the 13, and so my intuition tells me that it is probably necessary to prevent that damage by surrounding the 13 with other circles. (I'm at 7:55 in the video now). No matter what, the 13 has issues on every side, so it seems best if none of those issues are along the straight borders. The only exception would be if using the thirteen on the edge is able to pull multiple other pieces away from the edge such that the combined improvement in efficiency is greater than what the 13 itself may give. So Assessing this specific matter is a good foundation to start, then hone in on the solution once this matter is fully accounted for. To summarize, beginning with the larger pieces is an excellent idea, but it also needs to be coupled with some awareness of efficiency. Once the larger pieces are placed correctly, one needs only to correctly incorporate all the smaller pieces.

I love it when puzzles like this come out where i can see what is going on!

This puzzle looks very challenging. So happy to see that you are able to solve it.

Not having watched to the end yet, my instinct would be to use the small pieces in the corners and the larger pieces in the middle. It seems like you'd have less dead space between pieces and the walls/corners/other pieces that way.

I would start by dividing the pieces into the six largest and six smallest, and I would pair-together the largest of each group and second largest and so on to the smallest in each group. My process would be to notice how the numbers relate to the fits to improve my intuition on the mathematical balances until the balances make for the most even spatial distribution (balances plural, because the math will balance very different in the corners than in the middle, so generally there are two balancings to account for). When it's spatial distribution is as even as possible, that should be the same as having fit them into the smallest possible square, thus they should fit.

Jigsaw and packing-puzzles are my favorite.

when I saw that the square has the minimal dimensions to fit the disks, I thought about an optimal configuration the 1 could be in to minimize empty space, which is surrounded by four disks that are tangent to it and their other two neighbors. I was pleasantly surprised that the solution shown didn't use that, and the fact that some of the disks still had wiggle room inside the square! it's great when a puzzle defies expectations, because that's how something gets you to think even more about a solution.

This is the first puzzle ever that I feel like I really want to own!

One story that many people might have read. Once a professor walked in the class with an empty jar. He asked the students, is this jar empty? To which students replied, Yes! He took out a packet of large pebbles and filled the container. And asked, can I fill it anymore? The students replied no. He then took a packet of small pebbles and placed them in the jar. The small pebbles took the spaces between the larger stones and filled it. Professor asked the same question again, can I fill it anymore? The students replied no. He then took out a packet of sand and started to fill the jar by slowly shaking it. In few seconds, the sand took all the possible spaces left by the pebbles. The professor repeated his question. The answer was again, no. Now the professor took the water bottle and and poured it into the jar. The water now replaces all the air and filled the jar completely. I remembered this while watching your video 😇👍

Simple and tricky, I love it.

10:30 I'm loving the puzzle jam in this video!

Thanks for the entertaining video Mr. Puzzle!

In general circle packing problems are really hard, but the main thing to keep in mind is optimizing the gaps that end up being forced between adjacent pieces. So if you think about the circles, the smaller it is the bigger its curvature, and therefore the bigger the "angle" it makes with another one. At the same time, smaller circles have a lot less area, so you want to use them to fill the gaps. So if you think about it, you'll probably need the biggest pieces adjacent to one another, with one of the smallest pieces filling the adjacency gap, to have good gap economy with it.

I think you can calculate the answer instead of trial and error. For example, the sum of the diameters of 10, 8 & 11 = 18.6; 11. 2. 7 & 5 = 19.2; 12, 13 & 5 = 18.6 and 10, 6 & 12 = 18.2. They will not be exactly the same because there will be wastage.

Awesome puzzle!

Of course the engineer finds an algorithm to solve this puzzle! Great work as always Mr. Puzzle!

This is really interesting. I really hate most puzzles as I am no good at them. But I think I would REALLY like this one. I love conservation of space and I feel like I would find this enjoyable.

This is actually a clip I won't watch to the end (yet), have to get one myself ☺️

Such a wonderful puzzle. Wondering to make own one 🧐

My personal thinking pattern went as such: -Find the maximum amount of units that can fit against one side -Find the maximum amount of units that can fit through one diagonal (using some more complex maths or calipers) -use what we know to rule out all impossible combinations -remind yourself that you want the highest possible mean number of pieces along each side [COMPLETE!] [not complete but definitely one of the quicker ways to pair down your options]

2D packing puzzles like this seem so simple and straightforward at a glance, when in reality, they are some of the most difficult puzzles, because there is usually only one extremely precise solution and numerous near-solutions.

This is sooooo SATISFYING

My first instinct was, since the puzzle was designed on mathematical principles, the solution would be easier to figure out using similar mathematical principles, particularly those used in Sudoku (each side equals a certain number) but after thinking about it for a moment I realized two things - first, that's not much better than random "blind" guessing and second, the sum of the pieces would naturally and organically fall into a pattern of equaling the sides regardless as the fixed dimensions would force that outcome. Basically, the mathematical aspect is "baked in" to the solution whether you consciously use it or not.

And you could also use the circles as Tower of Hanoi pieces 😄

Nice work !

I can't help but feel your average school kid could do this in no time (silly maths problems), but that was great to watch. So frustrating as you kept getting closer and closer but to me this is a proper puzzle. No tricks, no traps just a solution that anybody could find in a number of ways. Thanks for sharing!

This was more entertaining than I thought it would be 😃

Interestingly, in the US wire gauge system, given a 1 in diameter, 16 gauge wire is sized so 16 pieces can fit in the circle, and for 12 gauge, which is a larger wire diameter, only 12 wires can fit. I’ve often liked the metric wire size system where the diameter of 1.5 mm is the actual wire size, and as the number increases so does the wire size; thus, this is much more logical.

This was so interesting! ...and also low-key sounded like a problem my AP Calc BC teacher would give us lmao I think it would be super cool to see you do some puzzle apps (like the room or house of davinci) because I find those super interesting and I'm curious as to how long it would take you as someone who has so much experience solving puzzles

To answer your question at 1:30: Let's say the radius of the small circle is 1 inch, which means its area is pi sq. in. That means the area of the 13-circle is 13pi sq. in., and so it's radius must be √13, and it's diameter must be 2√13, or about 7.2111 inches. Divide this into 18.59, and we get 2.578. Thus the square's side is 2.578 times the diameter of the biggest circle.

That's what I do with my fantasy Hosta Garden. Fit circles into square shape optimally. Cool puzzle.

Great job mr. Puzzle

Fascinating

Great Job.

I would start with the biggest pieces away from the edges. Looks hard.

13:30 - I got nearly as excited as you 👍

For a moment I thought a pattern would be 2 disks that touch added to 13... but there are a few that don’t.. great puzzle... I liked this

These always puts a smile on my face (which is hard to do) 🤣😁👍

Looking at the solution, if you add up the lengths of the tangent circles on each edge West- 29 North- 29 East- 29ish (that 5 isn't quite tangent) South- 25ish This might be a good clue for how to start optimizing. Look for a critical number to efficiently pack a side, then find different combinations of circles that make that number and apply them to those sides, with corner circles being used twice. I think this is the key for why you still need large circles in the corners, or else you'd be "doubling up" smaller numbers. But also why the 13 doesn't go in a corner: it's just too fat to justify and cuts down your combinations too much!!

Fun one!

The arrangement of the circles 11, 2, 7 and 5 is , 0.17% longer than the one with 11, 8, and 10. So either it's a rectangle or there is a very tiny gap between the latter.

Well done 👍✅

Are you saying PACKING puzzle? I've been watching for a year and every time I heard "PECKING" puzzle :D Both make sense

Excellent 😯

Should be able to fit roughly 2.5 of the 13x circles in the square. Area of 1x circle is 3.14 units. Area of 13x circle is 40.8 units. Diameter of 13x is 7.2 units. Roughly 2.57 13x circles can fit across the length of the square.

Want to try this one!

Bravo.

u've been on youtube for so many years damn, keep it up bro!

I was cheering you on. Got it --cool

You are so great Mr.Puzzle!! Thank you for another great video. Without using the term "dope" too! I keep trying to get your fellow youtube puzzler, Chris Ramsay, to stop saying that word, yet he still does. I commend you on not ever using that word. You are so entertaining in your videos and I appreciate it very much!! Peace&Love&Light ☮☯️🕉

Really tough puzzle but my OCD wants you to align up all the numbers so they have the correct forward facing orientation.

I think it would have went smoother if you tried rotating the pieces more!

Circle Packing Algorithm is godly here.

I'll leave 1, 2 and maybe 3 as wildcard pieces to fill gaps at the end. Starting with the biggest circles trying to cover as much area as possible. Edit: I see, that's what he did. Edit 2: Switching 13 with 12 and 3 with 4 leaves less free play, as I see.

I would have put in the 4 biggest with one on each side first, otherwise you lose too much space in the corners.

Im guessing start with smallest outside. That would lessen the side and corner spaces. Then adjust. But dont know. This might be my favorite puzzle on your show

I feel like with that kind of puzzle kind of like what you did towards the end is switch a lot with the number + or -

You need the sqrt of the numbers on the circles basically from 1 to 13, then you are left trying to make 3 or 4 add up to the edge size of the square making it much easier than randomly trying configurations, but defeats the point if it was to test spatial skills.

I just think it's nuts that you can even calculate the solution to this haha I wouldn't even know where to start

I'd have probably approached this by packing from the largest circle down, working on the assumption that the gaps for the small pieces will reveal themselves. Not sure how that would go :-)

Put them on the side, you even can put more than 13 !

I love your videos, thanks for making them!

Doing a little bit of algebra we can figure out how many times diameters of circle 13 one side of the square can fit: Friendly reminder: area of a circle = πr^2 We'll call any area A(n) (n being the circle #), any radius r(n) and diameter d(n) (same logic) A(1)=πr(1)^2 A(13)=πr(13)^2=13A(1) => r(13)=sqrt(13)r(1) d(13)=2r(13)=2sqrt(13)r(1) lets put all of that in relation with the square of side c=18.59r(1) c/d(13) = 18.59r(1)/2sqrt(13)r(1) =~2.58 We can fit the diameter of circle 13 2.58 times on the side of the square :) I skipped over a lot of steps so I apologize if it's not very understandable !

The solution ending up being drastically different from what predicted. I would have thought the smallest circles would be in the corners, to minimize the amount of space wasted there and allow for maximum packing. Instead, 3 of the 4 largest circles are in the corners.

The length of one of the squares edges should be 2.56 times the diameter of the largest circle. (18,59/2xsqrt13)

Since you know the relation between the size of the pieces, I would start by working from the outside in.

What about Square pieces that have to pack into a Circle?

2.57 of the diameter of circle 13

Well done. For entrees this evening, I have swordfish meatloaf with onion marmalade, rare roasted partridge breast in raspberry coulis with a sorrel timbale.

I would think the sums should be the same horiz, vertical, diagnal

Seeeeeeeeeeeeeeeeeeehr gut 🤩

The only logical way I can think of solving this is to take the area of the square and subtract the area of the circles to tell you how much negative space you will have, but I don't know how to measure the negative space using only circles.

18.59r = 2xa | 13r²π = a²π | r = 1 x = 18.59 / 2√13 = 2.5777969... => ~2.58 That was fun. Has been a while since I did some math just for fun.

Lovly problem, got a bit nerdy.. area of the square is 110.004 times the smallest circle and 345.59 times the radius of the smallest circle, and the biggest circle that can fit in the square is 86.4 times the smallest circle

I just got my first puzzels today.. 5 hanayama puzzels😊

Cool

Nice

I belieave the trick here is to get n°29 like:12/7/10 ? /13/? ?/9/ ? At 6:23 will fit this line.

The first packing problem that got a difficulty rating of 5! Let's goooo!

Just a fun fact, from your number: all 13 circles cover 82.72% of the square.

2.577969 diameters of biggest circle

Congrats to Risa 12 and I think it's about 2.5779691times diameter of biggest circle. Is this the only solution?

Mmm for reason I thought It'd end up being spiral shaped

Rated a 13 out of 13

Yeah I would just try placing the pieces in decreasing order. If you can't fit piece X, then there's no point in trying to place the pieces