BBC News has reported the new term SPOC in Next Big Thing. We shall remind the short story.

**Distant past**: bricks-and-mortar universities.

For hundreds of years students learned in so-called bricks-and-mortar universities. The general scheme includes

(1) paying large tuition fees for

(2) access to real professors and also

(3) enjoying social life away from parents

(4) hopefully gaining a degree at the end.

However the Internet is changing the education.

**Recent past**: MOOCs (Massive Open Online Courses).

Many students have probably heard the term MOOC, see more details in the Wikipedia article. The term MOOC was coined in 2008 and first 3 real MOOCs appeared in September 2011. By September 2013 there were dozens of MOOC providers in different countries. Initially MOOCs didn’t have the same factors as the conventional universities:

(1) large tuition fees

(2) real professors

(3) any social life

(4) certificates.

Several MOOCs have started to offer

(1) discussions and chats with instructors

(2) assessments with identities verified

(3) a small charge for certificates.

The main drawback of MOOCs has been the low retention rate: about 8% of all enrolled students actually pass a final test.

**Next Big Thing**: SPOCs (Small Private Online Courses).

It seems that term SPOC has just been coined by BBC on 24th September 2013. There is no Wikipedia article on SPOCs yet: checked on 27th September 2013. So we shall briefly describe the key points of the BBC report.

(1) Harvard says that we are already in the “post-MOOC” era.

(2) Access to SPOCs will be restricted to dozens or hundreds.

(3) Proper guidance will be provided for the selected students.

(4) Students will be more rigorously assessed than in MOOCs.

We have started our SPOCs (distance courses for maths candidates to top UK universities) in September 2010, which was 1 year before any MOOCs appeared and 3 years before Harvard started to think about SPOCs. We hope that other SPOCs (will) have the same key values as our current courses for top students preparing for MAT papers, Oxbridge interviews, STEP exams:

(1) quick and detailed feedback on regular written homework

(2) training on many problems harder than past exam questions

(3) best tips and advice from our recent successful students

(4) interactive quizzes offering gradual hints when needed.

Actually, we feel rather happy that big players like Harvard are catching up. The flexibility of small start-ups is a great advantage in tech innovations.

Small mammals are now ruling the world, while huge dinosaurs were wiped out. Here is the *powerful principle*: **the size does not matter, but the skills do**!

**Riddle 7**: which black area is larger in the picture in the top left corner

at the very beginning of this post: all small squares or two big squares?**How to submit**: to write your full answer, submit a comment.**Hint**: you may assume that each small black square has side 1.**Warning**: please mathematically justify your conclusion.**Restriction**: only the first correct answer will be rewarded.**Prize**: free 1-year access to one of our interactive web tutorials.**Update**: after Ariella’s attempt, the answer has been explained.

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sqroot(x^2) = abs(x)

since x^2 is positive (or zero) for all real x then sqroot(x^2) is also real,

sqroot(x^2) will give us the positive value only which if x was initially positive will be our origional value however if x was origionally negative we will now have it’s absolute value

Dear R S, thank for your comment. Next time you could reply to the post with the riddle you are solving. The current post finished with the poll: your personal best tip how to improve SPOCs or MOOCs. Yes, the square root of x^2 (x squared) for any real x equals the absolute value |x|, which is x for x>=0, but |x|=-x if x<0. To claim your prize, please self-register at our tutorials web site: click on “Create a New Account” and type the same e-mail you used to post the comment. Then e-mail blogger@master-maths.co.uk the title of the tutorial you would like to access. You could also try our free tutorial: after logging in, navigate to “Beyond Pythagoras’ theorem and applications”, click on “Enrol me in this course” in the left-hand side menu.

The small squares cover a larger area than the two large ones.

85 small black squares versus 84 small white squares

The two large black squares cover exactly 50% of the surface, since they occupy diagonal quartiles.

Dear Ariella, the claim “85 small black squares versus 84 small white squares” is correct, however our question was about the black areas, not white. Moreover, we hoped that you didn’t count 85 squares by hand. The first picture is a square 13 by 13 whose one side is occupied by 7 black and 6 white squares. We may count all black squares in the lattice 7 by 7 (including all external squares on the sides) and in the lattice 6 by 6 (interleaved with the already counted squares), so there are 7^2+6^2=85 black squares.

The claim “The two large black squares cover exactly 50% of the surface, since they occupy diagonal quartiles” can be correct only if the two large black squares are equal. However, they have sizes 7 and 6. Indeed, the left vertical side of the lower large square corresponds to 4 black squares and 3 white squares in the first picture. So the total area is 7^2+6^2=85 as in the first picture. Finally, the total black areas are the same in both pictures. If sizes were not clear, you can always ask us, because we reply within few hours in day time. Thank you for your attempt, we hope that you will solve more our riddles.