# How to solve hard problems: a key principle with examples

It would be much nicer to have a simple problem instead of a hard one. So a natural way to solve a hard mathematical problem is to make it simpler.

The key principle is to simplify.

At summer schools we often ask the simple question: how will you solve the equation $$(x-2)^2=1$$? Most students expand the brackets and then apply the quadratic formula.

The standard quadratic formula is a correct general recipe. However, a simple computer program will easily beat you. While you are solving one quadratic equation, the chip in your mobile phone can solve a billion of similar equations. So mathematics is not a cookbook of recipes, but a beautiful subject where efficiency, not brute force, is highly valued.

Efficiency often wins over brute force.

If we replace $$(x-2)^2=1$$ by a very similar equation $$(x^2-2x)^2=1$$, then expanding the brackets will produce a longer and more complicated equation of degree 4. Moreover, any computer program based on the standard quadratic formula will fail.

However, both equations $$(x-2)^2=1$$ and $$(x^2-2x)^2=1$$ have the same pattern: a square equals a number. Hence the equations can be solved in the same way. For instance, we could replace the expression under the square by a new variable.

A substitution is a common way to simplify.

In the equation $$(x-2)^2=1$$ we could set $$y=x-2$$. Then we get the simpler equation $$y^2=1$$, which has two roots $$y=\pm 1$$. Hence the original equation $$(x-2)^2=1$$ has two roots $$x=y+2=-1+2=1$$ and $$x=y+2=1+2=3$$.

Similarly, in the equation $$(x^2-2x)^2=1$$ we could set $$y=x^2-2x$$. Then we get the simpler equation $$y^2=1$$, $$y=\pm 1$$. Hence the original equation $$(x^2-2x)^2=1$$ is equivalent to the union of 2 simpler quadratic equations $$x^2-2x=-1$$ and $$x^2-2x=1$$.

How to simplify: try to keep a pattern.

Most students would probably solve the last two equations by using the standard quadratic formula again. We could actually continue the simplification and complete a simple square instead. Namely, $$x^2-2x=-1$$, $$x^2-2x+1=0$$, $$(x-1)^2=0$$, so $$x=1$$ is a double root. Similarly, $$x^2-2x=1$$, $$x^2-2x+1=2$$, $$(x-1)^2=2$$, $$x-1=\pm\sqrt{2}$$, $$x=1\pm\sqrt{2}$$.

There are many more complicated equations with the same pattern $$y^2=1$$. For instance, try to solve the equations $$(x^3-2x-1)^2=4$$,   $$(\sin x+\cos x)^2=1$$,   $$(e^{2x}-2e^x)^2=1$$.

• Riddle 3: simplify $$\sqrt{x^2}$$ for any real number x and explain your conclusion.
• How to submit: to write your full answer, simply submit a comment.
• Hint: the square root $$\sqrt{x}$$ of any positive real number $$x$$ is always positive.
• Warning: many 2nd year maths students at a top university get it wrong.
• Restriction: only the first fully correct public answer will be rewarded.
• Prize: free 1-year access to one of our interactive web tutorials.
• Update: R S has solved the riddle, read our confirmation.

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# How to enter top UK universities: 3 mistakes and 3 tips

We share a few anonymous stories how students succeeded or failed to enter top UK universities for maths degrees. We assume that our readers are familiar with our summary about top UK universities and maths entrance exams. This picture is only a rough guide:

Mistake 1: a late start in exam preparations.

Surprisingly many students ask us to help just about a week before their exams. It is good that these students try hard to submit as many homework problems from our distance courses as they can. It turns out that some of these late students are actually rather strong in mathematics. However, a week or a month seems insufficient, simply because learning deep maths concepts often takes months and even years. Many late students unfortunately fail.

Tip 1: try a similar real exam in advance.

The top UK universities for maths degrees (Cambridge, Oxford, Warwick, Imperial) usually give their candidates only one chance to gain a good grade at their entrance exams. For instance, Oxford selects candidates for interview in December by their results in a MAT paper that can be taken only in November.

We taught a strong Oxford candidate who couldn’t complete MAT for personal reasons that emerged on the exam day. However, the student gained a grade 1 in the harder STEP I exam earlier in June. So Oxford admissions tutors happily accepted this grade 1 in STEP I instead of MAT and invited the student to the interview. The student successfully entered Oxford.

Mistake 2: are personal statements important?

Another our student was preparing for a MAT paper, but submitted only few first homework problems from our course. About 2 weeks before the exam, the student e-mailed us their personal statement and asked for our opinion. The statement seemed brilliant, though we are not experts in assessing personal statements. However, by our past experience, a better progress in our course was needed to succeed in MAT. So we encouraged the student to focus on their MAT preparations. Unfortunately, it was a bit too late in this case.

Tip 2: focus on maths exam preparations.

University admissions tutors are often lecturers or professors who are pretty busy with their research and teaching. Moreover, mathematicians actually prefer an objective selection that is based on already gained results in real challenges. Shortly, you could outline your achievements (as a bullet point list) in maths competitions such as challenges by the UK Mathematics Trust. It seems worth spending much more time (95% or 99%) on training for proper challenges and entrance exams that bring long-lasting rewards.

Our students tell us about (and often thank us for) their success in the British Mathematics Olympiad, Senior Team Challenges and Senior Kangaroo Challenges despite we run distance courses only for entrance exams such as STEP, MAT papers and Oxbridge interviews. Smart students consider any serious olympiads or challenges as extra opportunities to practice problem-solving skills in non-standard situations beyond the school curriculum.

Mistake 3: over-estimating your own potential.

A few years ago one year 12 student gained a diploma in the maths competition at our summer school for maths candidates to top universities. Later the student applied to Oxford and decided to independently prepare for a MAT paper without taking our first distance course. Unfortunately, the application was unsuccessful. Then the student took a gap year, applied to Cambridge and completed our 4 courses for Oxbridge interviews and 3 STEP exams. This second attempt was better: a grade S in both STEP II and III, and a place at Cambridge.

Tip 3: learn from past mistakes of others.

Many people learn much more from mistakes than from successes (if it’s not fatally late). Watch this brilliant show by one over-optimistic hopeful who didn’t properly prepare for a challenge. If you would like to feel better after exams, hopefully you know what to do now.