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.

If you would like to share your useful experience with maths entrance exams, please leave a comment by clicking on “Leave a reply” above or e-mail Master Maths blogger.

**Riddle 2**: find the largest number consisting of only 3 symbols: 9, 9, 9.**How to submit**: to write your full answer, simply submit a comment.**Hint**: justify that your answer is the largest number among few alternatives.**Warning**: maths isn’t about recipes, but requires thinking “outside the box”.**Restriction**: only the first fully correct public answer will be rewarded.**Prize**: free 1-year access to one of our interactive web tutorials.**Update**: after attempts 1 and 2, the answer has been explained.

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9^9^9

Dear Ariella, thank you for your attempt with a correct idea. You have used two operations and should specify their order. Actually, TeX would not recognise “9^9^9” and will output the error: “Double superscript”. In TeX notations you could write [latex]9^{9^9}[/latex] or [latex](9^9)^9[/latex]. Please briefly justify which number is larger.

You could now self-register at our tutorials web site: simply click on “Create a New Account” and type the same e-mail you used to post the comment. After logging in, navigate to our free web tutorial “Beyond Pythagoras’ theorem and applications”, then click on “Enrol me in this course” in the left-hand side menu.

We hope that you will clarify your answer in a further comment and then we will enrol you on one of our tutorials.

[latex](9^9)^9[/latex]

Justification for this order is 9 to the power of 9 to the power of 9 .

Because a bigger number to the power of 9 makes a bigger number in total .

Dear Ariella, the correct answer is 9^{9^9}, not (9^9)^9=9^{9*9}=9^{81}. So first we compute the large number 9^9 and then take 9 to this large power 9^9. If we take the same large number 9^9 to the power 9 as in (9^9)^9, then we get only 9^{81} by the general rule (a^b)^c=a^{bc}. So the larger number is 9^{9^9}>9^{81}=(9^9)^9. We hope that you will correctly answer our future riddles. Thank you for your attempts!