#### 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!

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

##### MP3 Work together!

This is collaborative problem solving in which students discuss their strategies to solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

##### MP6 Be precise!

This is where our 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!

One of the things that the human brain does very well is identify pattern. We sometimes do this too well and identify patterns that don't really exist. 