The brilliant artwork in this puzzle was done for me by Mr. Cuddington. Here is the challenge. You want to print up a card game with one card of value 1, two cards of value 2, three cards of value 3… twelve cards of value 12…
Here is a page with 8 cards. Rules:

  1. Each page is printed a number of times equal to its lowest card. So this page will be printed up once.
  2. There can be no extra cards. Exactly one 1, two 2s, three 3s… twelve 12s.
  3. Each page must have the same number of cards. In this case you chose 8.
This page will be printed up four times (because 4 is the smallest number.) You can see that it will print up all the 4s, 8s and 12s that we need. It will also print up four of the 9s and four of the 11s that we need.
This page will be printed up five times (because 5 is the smallest number.) It will print up all the 5s and 10s that we need. It will also print up five of the 9s, but if you remember back – 9 was printed up four times on the previous page. 4+5 = 9. Yippee! Have we succeeded to print up all cards?
Maybe this is an easier way to decide. These rows represent the previous three pages…
Only six 7s were printed… and too many 11s were printed. We failed. See if you can succeed. Remember the rules:

  1. Each page is printed a number of times equal to its lowest card.
  2. There can be no extra cards. Exactly one 1, two 2s, three 3s… twelve 12s.
  3. Each page must have the same number of cards.
Having eight cards per page does not work. Why? Because the total number of cards printed is:

1+2+3+4+5+6+7+8+9+10+11+12 = 78

That’s not a multiple of eight. However 6 x 13 = 78. Can you find a solution with 6 cards? What about the other factors of 78… (2, 3, 13, 26, 39, 78?) Stop and try.

Here is a solution for six cards. This is page 1.
The second…
The third…
The fourth and last…

Again – let’s look at them all on one slide…

Do you think this is a unique answer?
No – there are many answers for four pages with six cards on each. Can you have five pages with six cards on each? Three pages with six cards on each?

Let’s add a scoring rule. Your score = Number of cards per page + Number of pages. Low scores are best. The previous solutions have 6 cards per page and 4 pages so their score is 6 + 4 = 10. Can you get lower?

This scores the same: 2 cards per page + 8 pages = 10.
This is the best I’ve found: 3 cards per page + 6 pages = 9. So we can score 9 for a deck of cards 1-12.

We should experiment with smaller and larger decks of cards of the type one 1, two 2s, three 3s… On the next three slides I’ll show the best results I’ve found for 1-13 and 1-14.

Again – for 1-13 there are lots of solutions like this. This scores 7 cards per page + 4 pages = 11.

Solutions like this 1-14 seem more difficult than 1-12 and 1-13 solutions, so this really is beautiful. The score is 5 cards per page + 5 pages = 10…

Another beautiful solution… The score is 3 cards per page + 7 pages = 10…

Did you notice that the score for 1-13 was higher than the score for 1-12 or 1-14? Obviously the score tends to get bigger as the size of the deck gets bigger, so this little hiccup at 13 is worth celebrating 😉

Enjoy this new puzzle – Email me any cool results you find.

12 Days

(MathPickle, 2018)
This puzzle is good for a wide range of grades. Younger students can focus on solving the impossible (8 cards per page) and possible (6 cards per page) basic puzzle. Older students who understand factoring can be given a freer hand to explore.

Download the pdf file here.

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. Try pairing up elementary students and getting older students to work in threes.

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 Use the right tools!

Students should use the right tools: 0-99 wall charts, graph paper, mathigon.org. etc.

MP6 Be precise!

Students learn to communicate using precise terminology. Students should not only use the precise terms of others but 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?

(http://www.corestandards.org/Math/Practice/)

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)