**Uncracked 33**

##### (Diophantus of Alexandria, c. 215-290)

Students working with exponents should be asked to find which values of n from 0-100 are possible by summing the cubes of three * positive* integers: a^3 + b^3 + c^3 = n.

Next, ask if the cubes can be * negative*. This is unsolved as you can see in the video, but do not tell your students this till the end of class! This video might be the best way to end the class – or to start the next class.

http://mathoverflow.net/ is a recommended resource for math entusiasts. It shows some of the history of this problem for discoveries n≤100:

*(1960s)*

*87=4271³–4126³–1972³*

*96=−15250³+13139³+10853³*

*91=83538³–67134³–65453³*

*80=−112969³+103532³+69241³*

*(1990s)*

*39=−159380³+134476³+117367³*

*75–435203231³+435203083³+4381159³*

*84=41639611³–41531726³–8241191³*

*(2000s)*

*30=2220422932³–2218888517³–283059965³*

*52=−61922712865³+23961292454³+60702901317³*

*74=−284650292555885³+66229832190556³+283450105697727³*

*For n≤1000, the problem is still open only for 33, 42, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, and 975 *

This problem also lends itself to algebraic exploration, but only for elite classes. As well as the algebraic exploration seen in the video check out Euler-Binet solutions 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. 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?