Number One!

This challenge is for everyone out there who loves their team to be #1. It’s a great number!

The number one is created using different positive integers in a fraction. The first time this is possible is when the maximum integer is six. Yippee! We could have made it (3×2)/6 or 6/(3×2)… it doesn’t matter. We could even have done (6×2)/(3×4).

We are always trying to include the most integers possible so in the last slide the (6×2)/(3×4) was great. When we go to a maximum of eight can we get to five integers?

No – this is not equal to the number one! We failed. Maybe it is possible to get 5 integers into our fraction with maximum 8 – but not like this.

By the way – let’s agree never to put a “1” in the top or bottom of the fraction. I’ll let you figure out why by yourselves if you’re interested. I’m not going to help you. 

Can you get to 8 integers with a maximum of 10? Think about how you’re going to figure it out before you go to the next slide. 

Of course, you could just use a calculator – but I’m going to be lazy and not type stuff in.

If it is going to work the prime factorization of the top and bottom must be identical. There are four twos in the top and four in the bottom – good.

There are two threes in the top and three in the bottom – good.

There is one five in the top and one in the bottom – good. That’s all the prime factors – so we are through.

Does this work?

If it does you can show it with a calculator or the lazy way of making sure the prime factors on the top and bottom are the same.

If it does not work find the smallest integer over ten that allows you to stick more than eight integers into the fraction.

No it doesn’t work. This fraction is not 1. Couting the factors of two top and bottom looks okay…

…but the threes are not equal. We failed to get to #1!

Try to mix up the integers in different ways – its impossible to get more than eight integers when the maximum is 12. 

How does this look?

Have we been successful?

This time we’ve failed for a different reason. We didn’t get as many integers as we could into the fraction. We can actually get 10 integers into a fraction with a maximum of 14. How?

Here is one way… There are many ways to solve these puzzles… I’m just showing you one way each time.

Now its time to split off into pairs and try to find the most integers that can be fit into a fraction with a maximum of 15, 16, 17, 18, 19, and 20. Students do not have to do these in any order.

Spoiler alert… answers up to a maximum 29 follow…

A fraction with a maximum of 15 can fit 11 integers.

A fraction with a maximum of 16 can fit 12 integers.

There is no improvement for 17, 18 or 19. The best you can do is the same as for a maximum of 16.

Once you allow a maximum of 20 there is a big leap! You can actually fit in 15 positive integers. I love how unexpected that leap was. 

The next increase is at 22. When the maximum is 22 you can fit in 16 positive integers. Keep going… when is the next increase 23, 24 or 25? (Of course, you might have guessed by now that a prime number like 23 is poison 😉

The next increase is at 24 – not 25. Do we next increase at 25 or 26 or neither?

I’m not going to show you the results for 25, 26, 27 – I’ll leave that for you to explore alone.

You can see this works for 28 because all the prime factors are equally distributed top and bottom.

The threes…

The fives…

The sevens…

The elevens…

Lastly – the thirteens…

That’s as far as I’ve solved it. It’s getting harder for me to do by hand, but maybe some of you are inspired to forge ahead 😉 

Please email me any cool discoveries:

Number One!

(MathPickle, 2019)

This group challenge is for students learning about prime numbers and prime factorization. Students will be working with fractions. 

I have solved the problem by hand up to 30, and would enjoy to hear from anyone who pushes the envelope further 😉



PS. The #1 photo used in this puzzle is by Bri Cibene.


Nick Baxter has written a computer program to find the optimal results. Here is the output. He got up to 150 before his computer became sluggish. Let’s look at one number far beyond what I got up to: 50. Here it is:

(3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 28) /

(2, 16, 26, 27, 30, 33, 34, 35, 36, 38, 40, 42, 44, 45, 46, 49, 50)

Equally interesting as the list of fractions is the sequence of the positive integers, n, that are improvements on n-1. Six (the first possible fraction) definitely offers an improvement on 5. Prime numbers never yield an improvement, but many composite numbers are also missing from the list.  Improvements are never separated by too much. 

Thank you Nick! That was such a beautiful set of fractions to receive 😉

Contact me: gord at mathpickle dot com, for any cool results you find.


Warmest thanks to Leen Busalih who is translating MathPickle into Arabic:

Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

MP1 Toughen up!

Students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP2 Think abstractly!

Students take problems and reformat them mathematically. This is helpful because mathematics lets them use powerful operations like addition.

MP3 Work together!

Students discuss their strategies to collaboratively solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

MP4 Model reality!

Students create a model that mimics the real world. Discoveries made by manipulating the model often hint at something in the real world.

MP5 Know the tools.

Students master the tools at their fingertips - whether it's a pencil or an online app. 

MP6 Be precise!

Students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

Students learn to identify patterns. This is one of the things that the human brain does very well. We sometimes even identify patterns that don't really exist 😉

MP8 Be lazy!?!

Students learn to seek for shortcuts. Why would you want to add the numbers one through a hundred if you can find an easier way to do it?


Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos 😉 For a free poster of MathPickle's ideas on elementary math education go here.

Gordon Hamilton

(MMath, PhD)