KiwiCo Crates Review – Are Kiwi Crates worth the money? For our family, yes. Here’s our KiwiCo review of Eureka Crate and Tinker Crate. These are great STEM kits for older kids.

As far as getting a subscription box for kids, I can’t imagine a better one, especially for tweens and teenagers.

‘Twas the week before Christmas, and I didn’t have a great gift idea for my tween and teenager.

I was on the email list for KiwiCo and was getting their holiday specials. All I knew was they offered fun and doable math, science, engineering, and art subscription kits.

Years prior, I saw a friend posting about her kids’ Tinker Crates and Eureka Crates. She was really happy with them. She said her teens loved them. Their grandparents bought the subscription for them each year.

Fast forward almost two years later, and I decided to look at what these Kiwi kit crates were all about. I ordered the subscription. (Updated: I wrote about how ideal these have been to do during the school closure and over summer break.)

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## Is Kiwi Crate worth the money?

For our family, yes. They are well-made STEM kits. The kids will actually enjoy using the project they complete. They will have the fun of building the kit as well as the fun of playing with or using it.

There are several great things that make Kiwi Crate worth the money. One of the best things is everything they need to build the kit is in the box. The directions are easy to understand and follow. I like that these are excellent projects for older kids as well. The projects are interesting.

While I can buy art supplies and other materials, it’s difficult to find STEM projects for older children. Yes, I can look online but then I have to get all of the materials. With subscription boxes, everything is inside. I cannot overstate enough how this is what makes it easy to say, “Get off electronics, and pull out your Kiwi Crates.” I don’t have run around setting everything up. They can take out their boxes and begin.

I love these Kiwi boxes because the kids look forward to getting them and completing them.

From the Eureka Crate, my son built a Mechanical Lock Box, a Ukulele, an Electronic Pencil Sharpener, a Desk Lamp, Stereo Headphones, and a Pinball Arcade game.

These are cool projects for a teen. Each was fun to do. He actually uses the items. He sometimes plays the arcade game. The kits are all different. Your kids will be proud they completed them. For the Eureka Crate and Tinker Crate, each kit will help teach about engineering and mechanical concepts as well as introduce new skills and reinforce existing skills and interests. The Eureka Crate in particular is fantastic for teens. The projects really held my teenager’s interest. Some of them had a lot of pieces but he took his time and completed them. Kiwi Crates are challenging enough for older kids but doable so they will stick with it and not lose interest.

Most of the Eureka Crates took my teen 50+ minutes to complete. (He completed the Stereo Headphones much faster but the Arcade game took over an hour.) These are worth it because they are such great projects whether or not your child is interested in STEM learning.

There are the easy-to-follow directions as well as supplemental materials if your child wants to pursue the concepts further. Kiwi Crates would work well for homeschooling. The student would have the hands-on building opportunity along with all of the supplement materials for extended at-home learning through additional experiments based on the concepts.

Also, when I compare the cost of a Kiwi subscription box to a summer camp or other class, Kiwi Crates are a great value.

## KiwiCo cost

*Will shipping make it cost prohibitive? *

Shipping is free in the United States. In addition, Kiwi Co. has discounts the longer you subscribe. Even at full price, the kits are a good value for what you receive.

## Subscription box for kids

For teens, this is what KiwiCo offers. We bought the Eureka Crate subscription and the Tinker Crate subscription.

**Eureka Crate: 14+:**Focuses on engineering and design**Maker Crate: 14+:**Focuses on art and design.**Tinker Crate: 9 – 16+:**Focuses on science and engineering.**Doodle Crate: 9 – 16+:**Focuses on creating and crafting.

I chose the crates that were more geared toward STEM and less on art because they sometimes do art at home (painting, drawing, etc.) whereas they didn’t have a way to do mechanical and engineering projects at home.

## KiwiCo Crates Review

While my kids have done a lot of STEM kits and other sets, they never had a kids subscription box. I had a lot of questions before deciding to try Kiwi Co.

## Would they build the kits?

*Would this be one more thing to pile up in the closet, untouched? *

They were doing their kits regularly in the beginning… within days of receiving them in the mail. However, after the third month, with school and activities, the crates started piling up.

However, the great thing is these kits were the perfect thing to do during the school closure and over the summer. We are renewing our subscription because they’ve enjoyed them so much.

*STEM kits are popular; what sets Kiwi crates apart? *

Kiwi Crate has kits for all ages, all the way through the teen years. They have kits in certain disciplines: Art, Geography, etc. but they all cover additional concepts. The Tinker Crate and Eureka Crate have working parts and are very detailed.

It’s difficult to find things for older kids to do to hold their interest but these KiwiCo crates do. They can take it to complete at grandma’s or do it at home. They won’t need to hunt around the house for additional materials. Everything kids need is included in the subscription box for them to complete the kit.

I also really like how these boxes are compact and stack nicely for storage.

## What is in a Kiwi Crate?

*Would my kids — who were definitely aging out of toys — want to do this? *

They were excited to do them when they received them. After a few months, the kits were piling up. But that’s not to say the kits weren’t fun. It’s just that my teens kept defaulting to electronics when they had free time. When they sit down to do the kits, they love them. They are proud they built the sets on their own. They enjoy using or playing with the kits when they’ve completed them.

*Will the kits be too easy? *

My 11 year old liked the Tinker Crate. It was on par with his skills. Some of the kits were just right; a few of them were a bit too easy. However, they were perfect for him to complete on his own without getting frustrated.

He was able to follow the directions, take his time, and be responsible with organizing the pieces. He was proud he accomplished the projects.

However, the sets weren’t “as cool” as the Eureka Crates. Now that he is 13, we will be renewing his subscription and getting the Eureka Crate.

*Is everything included in the kit? *

Yes, everything is included in each kid’s subscription box. This is wonderful. You won’t need to even get a pair of scissors. A few of the sets included a mini screwdriver.

Basically, I didn’t have to worry about: *Would these crates be another thing I have to nag my kids to do?*

Like many parents, I know I’m always looking for ways to engage my tween and teen and get them away from electronic devices.

I also know I’ve bought my fair share of science and engineering sets, art kits, and other enrichment sets and projects thinking they would do them but they never did.

Or the kids do them but it was a chore — not something to look forward to but something to cross off their list. All of this would just add to my overall guilt about wasting money and the pressure to offer my kids enriching experiences.

With Kiwi Co. subscription, we have the satisfaction of completing the kits, enrichment, and not wasting money.

## Kiwi crates as gifts

KiwiCo boxes make great gifts for kids of all ages. They have sales and discount coupons often. This subscription service is for all ages. Kiwi categorizes the kits by interest.

For wee ones this would be Discovery and Exploration and Playing and Learning.

For ages 5 and up, there is Art, Science, Design, Technology, Art, Geography, Engineering, and Math. I really liked they were broken down into such specific age ranges.

You may have heard of STEM-based learning or STEAM-based learning. This stands for Science, Technology, Engineering, A for Art, and Mathematics. After receiving several crates now, we learned that Kiwi crates combine many of these STEM/STEAM disciplines in their kits.

For our family, the best part is the Tinker Crate and Eureka Crate are appropriate for tweens and teens.

## Kiwi Crate vs LEGO

When I buying the KiwiCo subscription, I realized this was the first year we didn’t buy LEGO sets for my kids. I love LEGO bricks and sets and so do my kids; they’ve served us well.

However, to get a challenging kit for them now that they’re older, I need to spend hundreds of dollars. Then LEGO model will have to sit out somewhere — collecting dust — and we just don’t have room for it.

My kids love doing LEGO sets but I just can’t justify the expense for something that will take them several hours (at best) and then it’s done.

These Kiwi crates are fulfilling the same function as the big LEGO sets for us. They give my (older) kids:

- Something to build and construct
- Opportunity to follow directions
- Chance to be responsible and organized
- Opportunity for patience and follow through
- Learning experiences and enrichment
- Something to be proud of
- Something fun to do that isn’t electronics

## STEM-based learning

My kids are really into STEM. They are in STEM clubs at school, and my older child takes honors science in middle school. I was happy thinking I could buy the Kiwi Crates for my kids and they would enjoy putting them together.

What was also appealing is that everything would be in the kit. As much as it’s great to browse Pinterest for STEM and other creative ideas, it takes so much time to gather the supplies. The appeal of the Kiwi subscription is that everything would be there in one box.

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I’ll admit there was a big part of me that worried they wouldn’t enjoy them. I thought they would be curious and have the thrill of unboxing the item but then might lose interest.

However, that wasn’t the case. They look forward to their subscription boxes and building the kits.

One of my children asked for an electronic pencil sharpener — he’d talked about it for months. I saw a picture on the KiwiCo site with a child putting together a pencil sharpener! I thought that was really cool that they offered even practical kits as well. Imagine our surprise when that was the first kit!

## How much is a Kiwi Crate subscription?

To ensure you get the best price, look on their website for a base price. Then sign up for their emails. They always have specials going and Kiwi coupon codes. What’s also nice is when you do sign up to get a subscription, they offer you discounts to add on additional months.

## Ordering the Kiwi Crate

I ordered the Tinker Crate for my 11 year old and the Eureka Crate for my 13 year old. I didn’t want them both to have the same kit.

Being my older child is capable and good at following directions — and goodness knows they’ve both been doing LEGO sets for the majority of their lives — I opted for the oldest-age kit which is the Eureka Crate.

Kiwi runs discounts and specials often, and I was happy with the deal and price. Sometimes there are special codes, etc. We received the kits in three days, plenty of time for wrapping and to put under the tree.

## What I liked about Kiwi Co

First off, we ordered are kits to be sent to the same address. Both of them fit in a larger box. This box is decorated really nicely. I also appreciated it was the correct size — the perfect fit, really — so that it didn’t waste.

The eco-friendliness shipping box aside, the kits themselves come in fantastic boxes. What I love about them is they are made with sturdy cardboard. They are perfect for then storing the kits until your kids complete them.

They stack nicely in the closet. Most toys that my kids used to play with didn’t come with boxes. This meant it was difficult to store. I am a huge fan of boxes!

Each box had a label which said what the kit was. Imagine my surprise when my son’s kit was an electronic pencil sharpener!

My younger son’s kit said it was a Paint Spinner. He had one years earlier which he enjoyed, so both seemed like activities my kids would enjoy building and putting together.

## What I didn’t like about Kiwi Co.

This is a minor compliant but thing is that even though they asked for both of our kids’ names, on the shipping box, only one child’s name is on it.

So if you order more than one set to the same address, you will want to not wrap the shipping box and present it as a gift. However, because only one box came, I opened it up and saw both of the sets were in that box.

Also, think about if your kids will care whose name is on the shipping box. Mine don’t care.

## Giving Kiwi Crate as a gift

Kiwi Crates make great gifts whether you give them a subscription or as a single boxed kit, one time.

The box makes a nice presentation. Everything the child needs to complete the kit is inside. What’s great about this is there won’t be a burden on the parents to get additional supplies. This gift is all-inclusive.

In our case, because they were each in their own nice box, I was able to wrap these up easily. When our kids unwrapped them, they were interested and curious. Again, it’s difficult to surprise older kids, especially tweens and teens. For gifts, they usually get board games, t-shirts, some books, and yes, video games.

We love finding STEM activities for them. They both said they wanted to do their kits over winter break.

I knew they would eventually do them and would want to do them but the fact they came up with the idea on their own, instead of wanting to play a video game or watching Youtube, was surprising.

## Building Eureka Crate and Tinker Crate

Both kids enjoyed doing their kits. The first kits they received took about an hour. (Some of the future kits took a little longer which was great.) The directions were detailed but in an easy-to-follow way, not complicated and cluttered.

What I like is that it teaches kids patience and to follow directions. They learn to take their time and find the pieces they need — just like they did when building their LEGO sets. They followed through until they finished.

We ordered the 3-month subscription and renewed for another three months. In total, our kids enjoyed the subscription for six months. We’ve received six Tinker Crates and six Eureka Crates.

I wish we ordered for the 12 months but we had a few months where they didn’t do them. Our rule was we would stop after they started piling up. However, because they completed them over the summer, I should have kept the subscription going. Updated: I’m ordering Kiwi again.

## KiwiCo Crate Review

### Easy directions

Colorful and includes pictures of the pieces they need. These Design Booklets feature what you need to know plus additional information. As an example, in the Electronic Pencil Sharpener kit, there is more information about the limit switch and about pencil lead.

### Educational

Just like any well-made game or activity, these kits teach a lot. My kids are learning about mechanics, engineering, math, physics, and more. It’s educational to build the project and then fun to use it or play with it.

Kids build and refine fine motor skills as well.

We don’t home school, but I could see how it would be appropriate to use to supplement the school day.

There are also extended learning opportunities for further enrichment. (We didn’t pay extra for this; it was included in the crate’s cost.) In addition to the directions for the one project, there are also other projects you can do.

These kits cover several disciplines — engineering, art, math, science, technology as well as geography for some kits.

Children learn to follow directions; to trust the process and learn patience; to sort and keep track of the supplies; and to follow through and complete a project. Again, all the while they are learning, building, constructing, and having fun.

### Good variety of projects

Kiwi Co boxes are interesting and haven’t been similar. Sometimes the projects have been more fun to build and sometimes they’ve been more fun to play with. Both are a win!

These are the sets they’ve received:

## Eureka Crate

- Electronic Pencil Sharpener
- Wooden Ukulele
- Stereo Headphones
- Mechanical Lock Box
- Articulated Desk Lamp
- Pinball Machine

## Tinker Crate

- Spin Art
- Color-Changing LED Crystal
- Light-Up Planetarium
- Arcade Catapult
- Hydraulic Claw
- Glowing Pendulum

You also have the option to go to their website and purchase specific kits.

As an example, even if you order the Kiwi Crate, you can buy a kit from the Doodle Crate. You can purchase one month, three months, six months, or 12 months. The longer the subscription, the more they discount it.

### Kiwi Co. Eureka Crate and Tinker Crate great for older kids

The kits have been detailed enough to hold older children’s interest. They are great kits for tweens and teens when not many “toys” and “craft kits” excite them.

These STEM kits have kept my kids interested and engaged in the process. They sit down and work until they have completed the kit. It’s something that’s “doable” and not something they view as a chore to do.

What’s also great is they have choices for older kids. Again, as children age, there are less options for them.

## KiwiCo box subscriptions for tweens and teens

- Atlas Crate is up to 11 year olds with a focus on geography and cultures.
- Doodle Crate is 9 – 16 year olds with a focus on design and art.
- Tinker Crate is recommended for 9 – 16 year olds and focuses on engineering and science.
- You may want to get Eureka Crate for tweens and teens. We liked Tinker but my older tween really liked some of the Eureka Crate projects better.

- Eureka Crate is for 14+ and focuses on engineering and design.
- Maker Crate is for 14+ and focuses on art and design.

Kids will enjoy constructing and building something they can play with and use.

### There’s usually a sale

Whenever you subscribe, you will feel like you got a good deal and didn’t pay full price. They offer discounts regularly.

### Spend time together

It’s been nice having my kids sit and work on their Kiwi kits while I make dinner. It’s is a great way to spend time together.

### Kids look forward to their kits

We ordered the Kiwi Crate 3-month subscription for the gift. I told my kids they had to work on and complete the kit before they open the next month’s shipment. And if they didn’t have their kits done before the next month, we wouldn’t renew it for another three months.

So far, they’ve been motivated to do their crates.

### Practical and enriching kits

Kids construct projects they can use. We have the pencil sharpener sitting out by our homework area. My son still plays with his spin art kit. My other son clipped his desk lamp on his desk and uses it when studying. In addition, he has the mechanical lock box on his dresser.

### Convenience and zero stress

Another benefit to these kids subscription boxes is you feel like you are doing something to enrich your kids. When my kids were younger, I enjoyed going to Michaels, JoAnn’s, Walmart, Target, and Hobby Lobby to find craft and science kits.

But with working and busy lives, I don’t have time to do this. Plus, there isn’t a sense of urgency to do these projects. However, when they arrive in the mail each month, there is a bit of excitement that makes my kids more interested in it.

I am no longer wasting money and filling up closets and drawers with art supplies and craft ideas for rainy days.

### Comes with everything you need

In addition to coming with all the materials you need, appealing directions and Design Booklets, it comes with that great box to store everything in, and some of the sets include a notebook.

Some of the projects don’t fit back into the boxes. However, some are part of the box as in the Spin Art machine which uses the box as part of the design. The Hydraulic Claw does as well.

### Helps refine fine motor skills

Kids need patience and to take care as they handle wires and small parts. It’s great to build and refine fine motor skills.

### You can cancel at any time

If you know how to log onto your KiwiCo account, you can cancel the subscription or change kits or pause your subscription. It’s easy.

I also very much appreciated getting an email reminder that our 3-month subscriptions were coming due to auto-renew. More on this below.

## Experience with Kiwi Co customer service

My kids opened their kits four days after Christmas. My older son was excited to do the pencil sharpener in his Eureka Crate. In putting it together, one of the wires broke apart.

It was a Saturday, but I emailed them right away over what I assume was a very busy customer service time for them.

I included our order number, an explanation of the problem, and also included a picture.

In almost exactly 25 hours, on Sunday, a rep wrote back to say they will send out a replacement part right away. They shipped it out Monday, and we received a replacement piece early afternoon on Wednesday. We haven’t had any other issues.

## Automatically renewing Kiwi Co. subscription

We’ve all had times where we agreed to pay for something one time but then see recurring charges on our credit card statements. This was a concern I had before purchasing KiwiCo. Would they continue to charge my account?

Not only did I get a confirmation from KiwiCo that my account was scheduled to renew, when I went online, I had options to easily cancel, to change the kits we wanted, or to put the account on hold.

I let them auto-renew. When I later learned I paid full price, I emailed Kiwi to tell them I saw a code for a discount. They credited my credit card for the difference for both kits.

I’m a huge believer in rewarding loyalty. So while I wish KiwiCo would have given me a discount up front as one of their loyal, existing customers, I was happy it was easy for them to honor the sale price when I wrote them.

So the only complaint would be they should have an automatic discount for their ongoing, loyal customers instead of focusing only on the getting new customers.

## Kiwi Crate keeps innovating

It seems Kiwi Crate overall is always trying to create the best possible experience for the kids. Each kit we’ve received so far has been of the utmost quality. It’s not cheap or skimping in any way.

There is the main kit plus options for children – teens to take the concepts to the next level by doing even more activities.

## Kiwi Crate options

We’ve haven’t tried one of the kits for younger kids. We subscribed to the Eureka Crate and Tinker Crate. All of the Kiwi Co’s crate lines are:

- Panda Crate: 0 – 24 months old
- Koala Crate: 2 – 4 years old
- Kiwi Crate: 5 – 8 years old
- Atlas Crate: 6 – 11 years old
- Doodle Crate: 9 – 16+ years
- Tinker Crate: 9 – 16 years — We love this crate! Perfect for tweens! (However, 12 year olds may prefer Eureka Crate.)
- Eureka Crate: 14 – 104 years young — Excellent crate for teenagers!
- Maker Crate: 14 – 104 years

## Happy with KiwiCo Eureka Crate and Tinker Crate

While my kids love doing these crates, I love they are learning as they put the kits together. I love seeing how proud they are that they made whatever it is and that it works!

Most of all, in this day of electronics, I’m thrilled to have found something that my kids are interested in. They are actively engaged in building their kits. All the while, they are learning and having fun.

So to answer the question,

*Is KiwiCo worth the money?*

We absolutely say Yes! My older children enjoy building their crates and want to build the kits. It’s a unique gift for tweens and teens.

I appreciate I don’t have to research online for these types of STEM kits or look through them online and at hobby and craft stores. My kids have fun doing their crates, and in the end, they have made something they are proud of.

### KiwiCo Crates Review for school closure ideas

My kids enjoyed doing these projects when they came in the mail each month. However, we all know life gets busy. They each had three crates still to do.

I bought the subscription when my kids were 11 years old and 13 years old. Now they are 13 and 15. They still really enjoy these crates.

They were piled neatly in a closet. During March, the first month of school closures, I told my kids to get off their video games and figure something out.

They pulled out their Tinker and Eureka Crates! They had such fun putting them together. Even better, they were proud of the Kiwi sets they built.

Again, the directions are broken down into steps so they didn’t need any help. It prompted us to look on the Kiwi Co site to see if we could buy single kits or a three pack that would be interesting.

In addition to building the kits, each box includes a booklet for enrichment. We didn’t do them but they seem fantastic. You could use these for enriched learning to make even more of these sets.

## Learned to play the Kiwi Co Ukulele

Another benefit to having this time with the school closures is one of my sons learned to play the ukulele he made. In the Kiwi booklet there is a recommendation for a website. We ended up buying a subscription to it.

So in addition to putting together the ukulele — working on following directions, patience, engineering, mechanical design skills — he also learned music.

### Kits for teens for summer break

These Kiwi crates have exceeded our expectations, engaged our tween and teen, and have enriched them in many ways. Our boys were proud to complete them and have enjoyed using their kits as well.

That Kiwi Pencil Sharper still sits out in our dining area, and we use it often. My son loves his Mechanical Lock Box, and hides his candy in it. He also uses his Articulated Desk Lamp. He clamped it to the desk in his bedroom.

We have the Hydraulic Claw, Arcade Catapult, and Pinball Machine out at times and they play with them.

When kids get older, there isn’t always a lot they want to do. The Eureka Crates and Tinker Crates held their interest. It’s also motivated them to come up with some engineering projects out of recyclables around the house.

Please note, this is an honest KiwiCo review. We did not receive anything from KiwiCo and chose to write this Kiwi crate review after buying the kits and enjoying them so much. There are affiliate links in this post.

Category: Math Games

**The Monty Hall problem** is a probability puzzle based on the American television game show whose host was Monty Hall. The popular show was called Let’s Make a Deal.

Imagine you’re having a really great day, and you’re feeling very lucky so you decide to go on the show. Monty Hall picks you for one of the games. You see three doors.

These are the rules of the game: Behind two of these three doors are goats, and behind one door is a brand new car. (Let’s say you’re trying to get a car and not a goat.)

Let’s say you already chose one of them, for example the red one. The host now, who knows what’s behind the doors, opens the green doors which have a goat.

Now the host offers you the chance to change your decision. What do you do? Do you stick with the door you chose or do you change? It may seem odd, but the odds are not $ 50 – 50$. You should switch the door – by doing that you’ll have almost $ 67%$ chance of winning.

Why exactly should you change your decision?

First you have three choices. The chance of winning is $ 1 : 3$. If you stick with the door you chose your chance will remain $ 1 : 3$. This means that there must be $\frac{2}{3}$ chance the car is somewhere else. If we know that the car is not in the last door, this means that there is $\frac{2}{3}$ chance the car is behind the yellow door. It is not certain that the car will be behind the yellow doors, but there is twice as much chance he is behind the yellow door than it is behind the red door.

Now let’s imagine having 100 doors. Now you got yourself in a kind of a bad situation. You have only $\frac{1}{100}$ chance to get it right. Now what if the host opens 98 doors with all goats behind them? Now you are left with only 2 doors. This becomes a bit clearer. The chance you got it right remained $\frac{1}{100}$, but the chance that the right door is somewhere among the other 99 door is now concentrated on the door you didn’t pick.

Category: Interesting math

Learn about the magic square. During the 14th century, Europe was devastated by the plague. Since medicine in the Middle Ages was extremely basic, they had no power over this terrible disease. People were terrified because they believed that Black Death was punishment from God.

They resorted to less verified sources and massively became superstitious. Here came the idea that mathematics can help prevent diseases and keep the devil at bay.

Some mathematicians created so-called magic square – a square table of order n containing numbers from 1 to n2 so that the sum of numbers in each row, column and both diagonals is the same. Soon enough, magic squares were everywhere.

## How do you calculate a magic square?

We will show you how to calculate a magic square.

## How to construct the magic squares of any order?

Order of the magic square represents the number of its __rows or columns__.

Construction of magic squares is divided into two different procedures:

- One where its order is odd
- One where it’s even

First, we’ll show the construction of magic squares with odd order.

Start with an empty square and draw a grid in it. For example, we’ll make a magic square whose order is 5.

Divide every side on 5 parts and draw parallels to make a grid with 25 squares.

Now on every side of this square we’ll construct a “triangle” of 4 squares. You will see these in green.

Now, starting from the top left diagonal, draw numbers from 1 to 25 in every other diagonal (including the added “triangles”).

If the order is $ n$, your last diagonal should end with n2. Now we should simply fill in the gaps. You take the numbers from those triangles and translate them vertically or horizontally to the furthest free place. If the number is placed on the top triangle, you’ll move it down. If it’s on the left, you’ll move it right and so on.

And the final result is:

Now we’ll show construction of magic squares of even order.

It is impossible to construct a magic square whose order is equal to 2. We’ll show how to construct a magic square whose order is equal to 4.

Draw a grid 4 x 4 and color them like this:

Starting from the bottom right corner, write numbers from 1 to $ n_2$.

The numbers in the white cells are here to stay. But the numbers in the pink cells must be rearranged.

Now erase the number in the pink cells and start the counting again from the top right number corner.

Now, the numbers in the white cells will remain as they were, and pink cells will take the numbers from the counting. This is the final result:

Of course, you can test it out. The sum of all rows should be equal to the sum of all columns and the sum of diagonal numbers.

Category: Interesting math

**The number Pi** has been known for almost 4000 years. Pi is one of the most fascinating numbers.

Imagine: if you write down an alphabet and you give each letter a certain number, in some part of pi your whole future can be written. This is because pi is infinite and there is no sequence of numbers that is repeating. **Pi** is ratio of the circumference of a circle and its diameter. The ancient Babylonians gave very rough approximation to pi- they estimated it to 3. The first real calculations were done by Archimedes of Syracuse.

**Pi** is ratio of the circumference of a circle and its diameter. The ancient Babylonians gave very rough approximation to pi- they estimated it to 3. The first real calculations were done by Archimedes of Syracuse. He showed that pi is one number between $ 3 \frac{1}{4}$ and $3 \frac{10}{71}$. He approximated the area of a circle based on the area of a regular polygon inscribed within the circle and the area of a regular polygon within which the circle was circumscribed.

Through all history, people found pi interesting. They wanted to know pi to more decimals, but couldn’t. Then the great Indian mathematical genius – Srinivasa Ramanujan started observing pi. He discovered remarkable formula for computing pi:

But it still all remained as estimation.

Later on, using the computer Alexander J. Yee and Shiger Kondo claimed to have calculated number pi to 5 trillion places. The main computation took 90 days.

Now we use the standard approximation to $\pi – 3.14$. But what if we want to remember pi to more than two decimals? We’d want to write it down as a fraction. For this we’ll use continued fraction.

Remember that the continued fractions come in form:

Let’s first take that $\pi = 3.14159$.

First number will be the floor of the starting number:

$ \mid a_0 \mid = \mid 3.14159 \mid = 3$

Now we have to calculate the first help variable $b_1$:

$ 3.14159 = a_0 + \frac{1}{b_1} \rightarrow b1 = 7.0625$

Using the help variable we can calculate $a_1$:

$ a_1 = \mid b_1 \mid = \mid 7.0625 \mid = 7 \rightarrow a_1 = 7$

Now you simply continue the process.

$ b_1 = a_1 + \frac{1}{b_2} \rightarrow b_2 = 15.96424$

$ a_2 = \mid b_2 \mid = \mid 15.9642 \mid = 15\rightarrow a_2 = 15$

$ b_2 = a_2 + \frac{1}{b_3} \rightarrow b_3 = 1.037$

$ a_3 = \mid b3 \mid = \mid 1.037 \mid = 1\rightarrow a_3 = 1$

This calculation leads us to our fraction approximation of pi:

$\pi \approx [3, 7, 15, 1] = \frac{3 + 1}{7 + \frac{1}{15 + \frac{1}{2}}}$

Now you’ll simplify this fraction and get that:

$\pi \approx \frac{335}{113} = 3.141592$

Also, don’t forget to celebrate March 14 (3/14), **the international Pi Day**!

Category: Interesting math

Have you heard the true story of seven bridges of Konigsberg? The famous mathematician from the 18th century solved the enigma of crossing all bridges in one route. But, let’s start from scratch so we can get the bigger picture.

__Konigsberg (now Kaliningrad)__ was a name of a city in Prussia, Germany back in 18-th century, until 1946. (In World War 2 it was occupied by Soviet Union and changed a name in Kaliningrad). It is located in on the coast of Baltic Sea. Though the whole city flows a river Pregel. This river flows in such way that in the middle of it lies an island and after passing the island, the river divides into two parts.

People build seven bridges so they could easily and quickly move around the city. Since people walked through there daily, they started to wonder if they can walk around the whole town, cross all the bridges only once and still walk over every bridge. People were quite intrigued by this question. Bridge looks like this:

The different areas are connected by **seven bridges**. We’ll mark them with a, b, c, d, e, f and g.

Let’s say there are four friends A, B, C and D living in different areas. They all wanted to know can they walk through the whole town, visit all areas and cross every bridge exactly one time and come back to their home.

This problem was solved by famous Swiss mathematician **Leonhard Euler**. He was very respected mathematician so people trusted him with this problem.

He approached this problems by imagining areas of land separated by the river into points connected with bridges or curved lines. This is how he represented this problem:

Now that he drew this he saw that the walking through every bridge is equivalent to our drawing with a pencil without lifting it of of the paper.

From this exact problem the foundation of graph theory was developed. Euler also set few new definitions:

**A network** is a kind of diagram where we have one or more vertices connected with non-intersecting curves.

**Vertex** is called off if it has an odd number of arcs leading to it. If it has even number of arcs, it is called even.

**An Euler path** is a continuous path that uses every edge of a graph once and only once.

If a network has more than two odd vertices, it does not have an Euler path.

This is how Euler solved this problem. Since every vertex here has an odd number of edges, it does not have an Euler path.

Let’s try an example with some simpler stuff you know from the beginning you can draw in just one stroke.

In these examples, you can start from any point you want, you could still finish this path.

The more complicated our shapes are, the more difficult for us is to see can we draw it without lifting the pencil of of the paper.

If a character can be drawn in one stroke, we say that it has an Euler path.

Now let’s think just like Euler thought. We should see what kind of a relationship do the vertices and edges have with the ability to draw Euler path. We’ll sort out vertices according to the number of edges that go through that vertex.

We can see, that in these, simple diagrams that we determined they have Euler path, all the vertices are **even**.

In general, all diagrams whose vertices are even has Euler path.

Let’s start with adding some odd vertices. Here we can see that we have four even vertices and two odd. Try to draw Euler path in this diagram:

We can also draw Euler path on every diagram that has exactly two odd vertices.

Let’s add even more odd vertices. Try to find Euler path yourself on the following diagrams.

From this we get to the following conclusion.

*A network or a graph has an Euler path if the number of odd vertices is either zero or two.*

This is how Euler solved this problem. Since four vertices here has an odd number of edges, it does not have an Euler path.

From this exact problem the foundation of graph theory was developed.

## Introduction to Graphs

Every graph consists of the set of vertices and set of **edges** – connectors of **vertices**. If said otherwise, graph is final which means that we can count how many vertices and edges a certain graph has.

We say that a graph is simple if every two vertices are connected only by one edge and there are no loops. **Loops** are edges that connect vertex with itself.

A graph that looks like a broken line that connects points in a line is called **path**.

Closed path is called **cycle**.

Complete graph is a graph in which every two vertices are connected by one edge.

Graph is connected if every two vertices are connected by some path.

Connected path without cycle is called the **tree**.

Category: Interesting math

What are prime numbers exactly? **Prime numbers** are numbers that can be divided, without a remainder, only by itself and by 1. For example, first odd number is number 2. 2 can only be divided by 1 and 2. Second number is 3, third 5 and so on… How can you determine whether the number is prime or not? First you divide it by 2 and see if you have a remainder, if you don’t, your number is not a prime. If there is a remainder that means that that number is not divisible by 2, and then you try it with 3 and continue until you get to a number you get when you divide your starting number by two.

For example number 7 is a prime number, because it can be divided only by one and 7, but 4 is not a prime number because, other than 1 and 4, can also be divided by 2.

Zero and one are not considered prime numbers. This is because, by the definition, prime numbers have exactly two divisors.

## Sieve of Eratosthenes

What if you want to write down as many primes as you want? For the small primes, the most efficient way is the Sieve of Eratosthenes.

For example, if you’d like to find all the primes less than or equal to 30, first list the numbers from 2 to 30.

The first number is 2. 2 is a prime number. This means that we’ll leave them, and mark him blue. Cross out all of the multiples of the number 2, we’ll paint them gray.

Second one is number 3. This number is also a prime number, so we’ll again mark 3 with blue color, and cross out its multiples.

You continue this procedure. And in the end you’re left with:

That means that the prime numbers from 1 to 30 are: 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

By **the Fundamental Theorem of Arithmetic** every positive integer greater than one can be written uniquely as a product of primes, with the prime factors in the product written in order of decreasing size. Knowing this theorem, we know the most important use of primes. We can learn a lot about certain integer if we know its prime divisors. It’s just like when you have a big problem that you’d like to solve – it is in your best interest to divide it into smaller problems.

Category: Interesting math

**Euclid of Alexandria** was a Greek mathematician. He was called the “Father of Geometry”. He wrote one of the most influential works in the history of mathematics – __Elements__.

Although Elements are mostly about geometry, Elements also include number theory. In this lesson we’ll give a little insight of this great mathematicians mind and his influence in the number theory.

**The Euclidean Algorithm** is a method for finding the greatest common divisor of two integers. Before showing the exact algorithm first we should set few rules and notations.

If a|b and a|c, we say that a is the common divisor of numbers b and c.

If at least one of numbers b and c is different from zero, then there exist only finitely many common divisors of b and c. The greatest of them is called the greatest common divisor of b and c. GCD is denoted as (b, c).

If we have two numbers whose greatest common divisor is one, we say that those two numbers are relatively prime. For example 40 and 13 are relatively prime. We denote this as:

You also don’t necessarily have to observe only two numbers. More than two numbers can also be relatively prime. If we have numbers $ b_1, b_2, b_3, … ,bn$ whose only common divisor is 1, we say that those numbers are relatively prime. Also if every pair of those numbers is relatively prime we say that there are pairwise relatively prime.

*Lemma 1.* If $ b = aq + r$, $ 0 \le r < a$, $ a > 0$, then $(a, b) = (a, r)$.

Put in words, this lemma tells us that if we have a number b, that is divisible with number a with remainder r, then the greatest common divisor from a and b will be the same as the greatest common divisor of $ a$ and $ r$.

As an example we can take 6 and 4. We know that $ 6 = 4 * 1 + 2$. We can see now that $ (6, 4) = (6, 2) = 2$

Proof. Let’s say that $ (a, b) = d_1$, $ (a, r) = d_2$.

$ b = aq + r \rightarrow r = b – aq \rightarrow d_1$ | $ r$ .This means that d1 is the common divisor of a and r.

Since we got that $ d_1 | a$ and that $ d_2 | b$ we can conclude that $ d_1 \le d_2$.

$ b = aq + r \rightarrow d_2 | b$. This means that $ d_2$ is the common divisor of $ a$ and $ b$.

From here we got that $ d_2 | a$ and that $ d_2 | b$ which means that $ d_2 \le d_1$.

We now got that $ d_1 \le d_2$ and that $ d_2 \le d_1$ which can only mean that $ d_1 = d_2$.

## Algorithm

Let’s assume that with repeated application of the remainder theorem we got the following string of equalities:

$ r1 = r_2q_3 + r_1$, $ 0 < r_3 < r_2$

…

$ r(n – 2) = r(n – 1)qn + rn$, $ 0 < rn < r(n – 1)$

$ r(n – 1) = rnqn + 0$

The procedure ends when we get remainder equal to zero. The procedure has to end in finally many steps.

Lemma 1 implies that the greatest common divisor of a and b is equal to rn because:

This means that $ (a, b)$ is equal to the last non trivial remainder in the upper procedure.

This process for finding GCD is called the Euclidean algorithm.

Let’s see how this algorithm works in practice.

*Example* Find GCD of $ 53357$ and $ 547$.

First we’ll divide $ 53358$ with $ 548$ and write it down as a sum of a product and remainder.

$ 192 = 1 * 164 + 28$

$ 164 = 5 * 28 + 24$

$ 28 = 1 * 24 + 4$

$ 24 = 6 * 4 + 0$

Since 4 is the last non trivial remainder this means that $(53358, 548) = 4$

Category: Interesting math

The **Koch Snowflake** was created by the Swedish mathematician Niels Fabian Helge von Koch. In one of his paper he used the Koch Snowflake to show that is possible to have figures that are continues everywhere but differentiable nowhere. Koch snowflake, curve or island is one of the earliest fractal curves that have been described.

How can you generate Koch snowflake yourself? First draw an isosceles triangle. And divide its sides on three equal parts.

Now, construct isosceles triangles whose one side is one middle third.

Take a look at the character we got. Now you can easily locate six new triangles. Again, you divide its sides on three equal parts and construct new isosceles triangles.

From here you just continue the procedure.

Category: Interesting math

**Gabriel’s horn** or **Torricelli’s trumpet** is the surface of revolution of the function $ f(x) = \frac{1}{x}$ about the x – axis for $ x \ge 1$. What is this exactly? First draw your axes and draw function $\frac{1}{x}$ for $ x \ge 1$.

Now you imagine it rotating around x – axis.

This figure has finite volume but infinite surface area.

How is this possible?

Well let’s try to find out what’s going on by calculation. This requires simple integral calculations. If we cut this horn into tiny regular slices we’ll always get circles with radius $\frac{1}{x}$ which means that volume of one little slice is equal to $\frac{1}{x^2} \pi$. And now that we know that we can find the volume of whole horn. Since the upper border of integral is infinity we have to use limit to get what we want.

This means that the volume of this trumpet is equal to π cubic units.

To calculate surface we’ll use surface integrals of second kind.

What confused people for a long time is a paradox that using this knowledge of its surface and volume you could fill it with a bucket of paint but the same volume of paint would not be enough to paint its surface.

This paradox is resolved because the surface we generated has no thickness and you can’t find any real-life objects with no thickness which you could paint. Paint itself has finite thickness bounded by the radius of an atom.

## Gabriel’s horn

Another thing you can think about is if you ever find real- life version of Gabriel’s horn or Gabriel’s trumpet could you play it?

Well no, since this trumpet is infinitely long, it will take you infinitely many years to come to its end, and even if you’re feeling especially adventurous and reach its end, it would be infinitely small so you couldn’t blow in it.

Category: Interesting math

**The Mandelbrot set** is named after Benoit Mandelbrot. He is a Polish – born, French and American mathematician.

He is largely responsible for the present interest in fractal geometry.

From his dedicated study of fractals came the great Mandelbrot set.

Fractals are never – ending, infinitely complex repeating patterns. The Mandelbrot set is a collection of numbers that are different from the real numbers you see in your every day life. These all patterns are happening in the set of complex numbers.

Every complex number has two parts- the real part and the imaginary part. Every complex number can be written as

Where the number i is the imaginary unit and $\imath_2 = – 1$.

This notation is very convenient because from this form we can easily find this numbers place in a complex plane.

So, how do we get to Mandelbrot set from here?

Let’s take some complex number b and let’s associate with it the following function.

We are interested in the behavior of our function in zero and then in that result and then in the next result and so on.

Now we’re observing how this function will behave if we take $ b = 1$.

The function value in zero is equal to 1, so we’ll observe how it acts in 1 and continue the process with the following results.

$ f_1(5) = 5^2 + 1 = 26$

…

The Mandelbrot set observes what happens to the size of these numbers. This size is the distance of the given number in a complex plane and the origin.

There are two options. The first option is that the distance from zero of the sequence gets very, very large which means it blows up – the size of the number goes to the infinity. The second option is that the distance is bounded- never gets larger than two.

Let’s observe the process we did. We got numbers $ 1, 2, 5, 26$ which means they are getting larger and larger and will continue to do so. This means that we’re talking about the first option.

Now let’s take a look what happens when $ b = -1$. We’ll always be getting numbers zero or $ -1$. This means that this is the second case.

Mandelbrot set is the set of complex numbers in which case two occur.

So how can you draw all these patterns? You just start iterating zero under $ z_2 + c$, if it takes a long time to get bit, you give it one color, and if it gets big very quickly you give it other color. This is interesting because you can’t predict how your function will act if you move change c just a little bit.

If we take fixed value of c, and continue observing this polynomial as we observed ones in Mandelbrot set we’ll get something called the **Filled Julia set**.

This means that the Filled Julia set is the set of complex numbers z so that under iteration by fc the values don’t blow up.

Category: Interesting math