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Puzzles: 100
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024Differentiate f(x) = x to the power x, and find where it reaches its minimum for positive x.Mental maths and number senseCoreScotiabankToronto · 2026
Try it first
What is the derivative of x to the x?
Show the worked solution
f'(x) = x to the x times (ln x + 1), and the minimum is at x = 1/e, about 0.368, where f is about 0.692. Take logs: ln f = x ln x. Differentiating, f'/f = ln x + 1, so f' = x to the x (ln x + 1). The derivative is zero when ln x = -1, that is x = 1/e, negative before it and positive after, so this is a minimum.
Why do both standard rules fail?
The power rule, n x to the (n - 1), treats the exponent as fixed; the exponential rule, a to the x times ln a, treats the base as fixed. In x to the x both the base and the exponent move, so neither rule applies on its own, and each one gives half of the right answer. Indeed the correct derivative is the sum of the two: x times x to the (x - 1), which is x to the x, plus x to the x ln x. That sum is a quick check on your final answer.
The curve x to the x falls from near 1 at zero to a minimum of 0.692 at x = 1/e and then rises to 4 at x = 2, and its logarithm x ln x has its minimum at the same point, which is why taking logs first is safe.
How does taking logs make it routine?
Think of converting a messy multiplication into addition before doing it, the way a slide rule does. Write ln f = x ln x; the right-hand side is a product of two simple functions, and the product rule gives ln x + x times 1/x = ln x + 1. The left side differentiates to f'/f by the chain rule, so f' = f (ln x + 1). This is logarithmic differentiationDifferentiating the logarithm of a function instead of the function, then multiplying back; useful when the variable sits in an exponent., and it works for any function of the form g(x) to the h(x).
The relationship
$f(x) = e^{x\ln x} \;\Rightarrow\; f'(x) = e^{x\ln x}(\ln x + 1) = x^x(\ln x + 1) = 0 \iff x = e^{-1} \approx 0.368$
e^{x ln x} x to the x rewritten with a fixed base
ln x + 1 the derivative of x ln x
e^{-1} where ln x = -1, the minimum
What it says in wordsRewrite with base e, differentiate the exponent, and set it to zero.
How do you confirm it is a minimum and state the value?
Check the sign of ln x + 1, since x to the x is always positive. For x below 1/e, ln x is below -1 and the slope is negative; above 1/e it is positive, so the function falls and then rises: a minimum. The value is (1/e) to the (1/e) = e to the (-1/e), about 0.692. Also say what happens at the edges: as x shrinks towards zero, x ln x tends to zero, so x to the x tends to 1, and at x = 1 it is exactly 1 again.
Where candidates lose it
The fast wrong answer applies the power rule, x times x to the (x - 1), which is just x to the x. It treats the exponent as a constant, and candidates who give it usually do so in the first three seconds.
The second loss is finding x = 1/e and stopping. The question asks for the minimum, so check the sign change and give the value, e to the (-1/e), about 0.692, together with the behaviour near zero.
What the interviewer asks next
Differentiate x to the (x to the x).
What is the limit of x to the x as x approaches 0 from above, and why?
Which is larger, e to the pi or pi to the e, and how does x to the (1/x) settle it?
Asked at Scotiabank, Quant, Toronto, 2026 (Wall Street Oasis): technical questions covering calculus (including derivatives of standard functions)
049Which is larger, e to the power pi or pi to the power e? Prove it without a calculator.Mental maths and number senseCoreQuant researchQuant trading
Try it first
Which way does it go?
Show the worked solution
e^pi is larger: about 23.14 against 22.46 for pi^e. Take logs of both and divide by e x pi, which turns the question into comparing ln e / e with ln pi / pi. The function ln x / x rises up to x = e and falls after it, so its value at e beats its value at any other number, pi included. Undo the steps and the order holds.
How do you turn two awkward powers into one comparison?
When two people race on different tracks, you compare them by converting to the same distance. Here the base and the exponent both differ, so convert each number into a common form. Take logarithms and divide by e x pi: e^pi against pi^e becomes ln e / e against ln pi / pi, the same function evaluated at two points. Logs and division by a positive number both preserve order, so whichever side wins the new comparison wins the original.
The relationship
$e^{\pi} \; ? \; \pi^{e} \iff \pi \ln e \; ? \; e \ln \pi \iff \frac{\ln e}{e} \; ? \; \frac{\ln \pi}{\pi}$
? the unknown direction of the inequality, the same at every step
\ln x / x the function whose largest value settles the question
What it says in wordsTaking logs and dividing by e times pi turns the question into one function compared at e and at pi.
Why is ln x / x largest at e?
Differentiate: the derivative of ln x / x is (1 - ln x) / x squared. It is positive while ln x is below 1 and negative once ln x passes 1, so the function climbs until x = e and falls after. Its largest value, 1/e, occurs only at x = e, so ln pi / pi must be smaller, and therefore e^pi beats pi^e. Equivalently, x^(1/x) peaks at e with value 1.4447, while pi^(1/pi) is 1.4396.
The curve x to the power 1/x reaches its maximum of 1.4447 at x = e and has already fallen to 1.4396 at x = pi, so raising both values to the power e times pi gives e^pi = 23.14, larger than pi^e = 22.46.
There is a second proof that needs no calculus beyond one inequality. For any x other than zero, e^x is greater than 1 + x, because the exponential curve lies above its tangent line at zero. Put x = pi/e - 1, about 0.156: then e^(pi/e - 1) is greater than pi/e, so e^(pi/e) is greater than pi, and raising both to the power e gives e^pi greater than pi^e. Give that one if the interviewer asks for a proof without derivatives.
Notice how close the race is: the two values of x^(1/x) differ by only 0.0050, because pi sits near the flat top of the curve. That is why rough estimation is risky here and a proof is needed: 23.14 and 22.46 differ by about 3%. The same argument settles a whole family: for any two numbers a and b with e at most a, and a less than b, a^b is greater than b^a, which is why 3^4 = 81 beats 4^3 = 64.
Where candidates lose it
The common loss is answering pi^e because pi is the bigger base, or trying to estimate both numbers to a decimal and getting lost in the arithmetic. The gap is only about 3%, so mental estimates can land either way.
The second is proving it backwards: assuming the answer and manipulating until something true appears, without checking each step preserves the inequality. Say out loud that taking logs and dividing by the positive e x pi keep the order.
What the interviewer asks next
Which is larger, 2^3 or 3^2, and why does the argument not apply to 2 and 4?
Find all pairs of distinct positive integers with a^b = b^a.
Which is larger, 99^100 or 100^99?
061A Rs 1,000 crore fund charges a 2% management fee and 20% of gains, with the performance fee taken on the gain left after the management fee. In a year with a 10% gross return, the manager cuts the management fee to 1%. What performance fee keeps the manager's revenue unchanged?Mental maths and number senseCoreTwo SigmaNew York · 2026
Try it first
Which performance fee keeps revenue at the old level?
Show the worked solution
About 28.9%, roughly 29%. Under 2 and 20 the manager earns Rs 20 crore of management fee plus 20% of the remaining Rs 80 crore gain, Rs 36 crore in all. At 1% the management fee is Rs 10 crore and the gain left is Rs 90 crore, so the performance fee must bring in Rs 26 crore: 26 / 90 = 28.9%. It only balances at a 10% return.
Why is the answer not simply 30%?
Think of a shopkeeper who takes a fixed rent from a stall plus a fifth of whatever the stall-holder clears after paying that rent. If the rent is halved, the stall-holder clears more, and the shopkeeper's fifth is a fifth of a larger amount. Cutting the management fee enlarges the gain on which the performance fee is charged, so part of the lost management revenue comes back automatically. Here the old fee of 20% on Rs 90 crore earns Rs 18 crore, up from Rs 16 crore, and the true shortfall is Rs 8 crore, not Rs 10 crore. Adding 10 points to get 30% repays the shortfall twice over in part.
At a 10% gross return the old terms earn Rs 36 crore, the fee cut alone earns Rs 28 crore, and a performance fee of 28.9% restores Rs 36 crore; at any other return the two sets of terms pay differently, crossing only at 10%.
How do you set it up in one line?
Write revenue as management fee plus performance rate times the gain after the management fee, and hold it equal. The new rate is the old revenue less the new management fee, divided by the new fee base: (36 - 10) / 90 = 28.9%. The investor's position is the mirror image: gross gain Rs 100 crore less Rs 36 crore of fees leaves Rs 64 crore, a 6.4% net return under either set of terms, which is a good check that you have kept the total fixed.
The relationship
$m_0 A + p_0 (G - m_0 A) = m_1 A + p_1 (G - m_1 A) \;\Rightarrow\; p_1 = \frac{20 + 0.2 \times 80 - 10}{90} = \frac{26}{90} \approx 28.9\%$
A assets under management, Rs 1,000 crore
G the gross gain, 10% of A, Rs 100 crore
m0, m1 the old and new management fee rates, 2% and 1%
p0, p1 the old and new performance fee rates
What it says in wordsHold total revenue fixed and solve for the performance rate on the new, larger fee base.
What does the swap change, even when revenue matches?
The match holds at one return only. Swapping management fee for performance fee moves risk onto the manager: less is guaranteed, more depends on the year. At a 5% gross return the old terms pay Rs 26 crore and the new ones Rs 21.6 crore; at 20% the old terms pay Rs 56 crore and the new ones Rs 64.9 crore. State the convention too: if the performance fee were charged on the gross gain instead, the old revenue is Rs 40 crore and the new rate comes to exactly 30%. Say which convention you assume before giving a number.
Where candidates lose it
The fast wrong answers are 40%, from doubling the performance fee because the management fee halved, and 30%, from adding back ten points. Both forget that the performance fee is charged on a base the management fee itself shrinks.
The second loss is giving one number without the conditions. A fee trade-off is always at an assumed return and an assumed fee base; say both and offer the sensitivity, because the interviewer's next question is what happens in a bad year.
What the interviewer asks next
At what gross return would the manager prefer the new terms to the old?
How does a hurdle rate of 5% change the new performance fee?
Why might investors prefer 1 and 29 to 2 and 20 even though revenue matches at 10%?
Asked at Two Sigma, Equity Capital Markets, New York, 2026 (Wall Street Oasis): the question was regarding how hedge funds operate, the 2/20 rule, and if one part of this equation changed
073Use Newton's method to find the square root of 2, starting from 1.5. How many correct digits do you have after each step, and why?Mental maths and number senseCoreQuant researchDesk quant
Try it first
Starting from 1.5, about how many correct digits after three Newton steps?
Show the worked solution
About 1, 3, 6 and 12 correct digits: the count roughly doubles each step. The update is x(next) = (x + 2/x)/2: 1.5 gives 17/12 = 1.41667, then 577/408 = 1.4142157, then 665857/470832 = 1.41421356237469. Each new error is about the old error squared divided by 2x, so if the error is 10^-k, the next is about 10^-2k. That is quadratic convergence.
Where does the update rule come from?
Think of guessing a side of a square room whose area is 2. If your guess is too big, 2 divided by your guess is too small, and the truth sits between the two. Newton's method for x squared minus 2 is exactly that: replace x with the average of x and 2/x. Formally, Newton follows the tangent of f(x) = x^2 - 2 down to zero, x - f(x)/f'(x) = x - (x^2 - 2)/(2x), which simplifies to (x + 2/x)/2. The averaging form is the one to use in your head.
Starting from 1.5, Newton's iterates for the square root of 2 have 1, 3, 6 and then 12 correct digits, doubling at each step because each new error is roughly the square of the old one.
How do you get the iterates without a calculator?
Keep fractions. From 3/2, the next value is (3/2 + 4/3)/2 = 17/12, then (17/12 + 24/17)/2 = 577/408, and the pattern continues: if x = p/q, the next is (p^2 + 2q^2)/(2pq). 17/12 is 1.41667, already right to 1.41. 577/408 is 1.4142157 against 1.4142136, right to 1.41421. The third step's fraction, 665857/470832, is too big for mental division, but you can predict its accuracy without doing it, which is the point of the question.
The relationship
$x_{n+1} = \tfrac12\Big(x_n + \frac{2}{x_n}\Big) \qquad x_{n+1} - \sqrt2 = \frac{(x_n - \sqrt2)^2}{2x_n}$
x_n the current estimate of the square root of 2
x_n - sqrt 2 the error of the current estimate
2 x_n about 2.8 near the root, so the new error is about a third of the old error squared
What it says in wordsEach new error is the old error squared, divided by about 2.8, so the correct digits roughly double.
Why is it the digits that double, and when does that fail?
Subtract the root from the update and the algebra collapses to (x - root 2)^2 / 2x. Squaring an error of 10^-3 gives 10^-6, so each step doubles the number of correct digits once you are close. The errors here run about 0.09, 0.0025, 2 x 10^-6 and 1.6 x 10^-12. The doubling needs a good start and a simple root: far from the root, or where the slope is zero, Newton can creep or jump away. Bisection, by contrast, gains one binary digit per step whatever happens, which is why desk code often brackets with bisection and finishes with Newton when solving for implied volatility.
Where candidates lose it
The trap is guessing linear progress, one or two digits a step, because that is how most iterative methods feel. Newton is special near a simple root, and the interviewer wants the word quadratic and the reason for it.
The second loss is getting lost in decimals. Work in fractions, 3/2, 17/12, 577/408, and state the error-squared rule instead of computing the third step.
What the interviewer asks next
Write Newton's update for the cube root of 10, and start it from 2.
Why does Newton converge only linearly at a double root?
How would you use Newton's method to find an implied volatility, and what can go wrong?
097How many integers from 1 to 1,000 share no common factor with 1,000 other than 1?Mental maths and number senseCoreQuant researchQuant trading
Try it first
Pick the count.
Show the worked solution
400. Since 1,000 = 2^3 x 5^3, a number shares a factor with 1,000 exactly when it is divisible by 2 or by 5. There are 500 multiples of 2 and 200 of 5, but the 100 multiples of 10 sit in both lists, so 600 numbers share a factor and 400 do not. Euler's formula agrees: 1,000 x 1/2 x 4/5 = 400.
Which numbers share a factor with 1,000?
Picture a hall of 1,000 people where everyone wearing a red badge or a blue badge is asked to leave. To count who stays, you need the red-badge count, the blue-badge count, and how many wear both, because they would otherwise be counted out twice. Write 1,000 as 2^3 x 5^3: a number shares a factor with it exactly when it is divisible by 2 or by 5, so only two badges matter, and the powers 3 do not add any new conditions. Every multiple of 4 or 8 is already a multiple of 2, and every multiple of 25 or 125 is already a multiple of 5.
Of the numbers 1 to 1,000, 500 are multiples of 2 and 200 are multiples of 5, with 100 multiples of 10 in both, so 600 share a factor with 1,000 and 400 lie outside both circles; Euler's product 1,000 x 1/2 x 4/5 gives the same 400.
The relationship
$\varphi(1000) = 1000 - \left\lfloor\tfrac{1000}{2}\right\rfloor - \left\lfloor\tfrac{1000}{5}\right\rfloor + \left\lfloor\tfrac{1000}{10}\right\rfloor = 1000\left(1-\tfrac12\right)\left(1-\tfrac15\right) = 400$
phi(1000) Euler's totient: how many of 1 to 1,000 share no factor with 1,000
1000/2, 1000/5 the counts of multiples of 2 and of 5
1000/10 the multiples of both, added back once
What it says in wordsRemove the multiples of each prime, add back the multiples of both, and you get the same answer as multiplying by the share that survives each prime.
Why does the quick product formula work here?
Half of all numbers are odd, and among those, four in five are not multiples of 5. Because 1,000 is a multiple of 10, the numbers 1 to 1,000 contain exactly 100 full blocks of ten, and in each block exactly 4 numbers, 1, 3, 7 and 9, survive both tests, so 100 x 4 = 400. The strip of 1 to 20 in the figure shows the pattern repeating, 8 survivors in 20. The product is exact only when the range is a whole number of such blocks: for 1 to 1,234 it gives 493.6, while a direct count gives 494.
Where does a question like this lead in an interview?
Usually to powers and remainders. Euler's theorem says a number coprime to n, raised to the power phi(n), leaves remainder 1 when divided by n, and that is the engine behind last-digit puzzles: phi(100) = 40, so 3^40 ends in 01 and so does 3^400. It also leads to probability: the chance that two large random integers share no factor tends to 6/pi^2, about 0.608. The habit the question tests is factorising first: once you see only the primes 2 and 5 matter, a counting question becomes a two-circle Venn diagram.
Where candidates lose it
The usual slip is 1,000 - 500 - 200 = 300, subtracting both lists and forgetting that the multiples of 10 were removed twice. Add them back once and the answer is 400.
The second loss is treating each prime power as a new condition, subtracting multiples of 4, 8, 25 and 125 as well. Every multiple of 4 is already a multiple of 2; only the distinct primes matter.
What the interviewer asks next
How many integers from 1 to 1,000 share no factor with 360?
What are the last two digits of 3^400?
What is the probability that two randomly chosen integers share no common factor?

e^{x ln x}	x to the x rewritten with a fixed base
ln x + 1	the derivative of x ln x
e^{-1}	where ln x = -1, the minimum

?	the unknown direction of the inequality, the same at every step
\ln x / x	the function whose largest value settles the question

A	assets under management, Rs 1,000 crore
G	the gross gain, 10% of A, Rs 100 crore
m0, m1	the old and new management fee rates, 2% and 1%
p0, p1	the old and new performance fee rates

x_n	the current estimate of the square root of 2
x_n - sqrt 2	the error of the current estimate
2 x_n	about 2.8 near the root, so the new error is about a third of the old error squared

Try it first

Why do both standard rules fail?

How does taking logs make it routine?

How do you confirm it is a minimum and state the value?

Where candidates lose it

What the interviewer asks next

Try it first

How do you turn two awkward powers into one comparison?

Why is ln x / x largest at e?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer not simply 30%?

How do you set it up in one line?

What does the swap change, even when revenue matches?

Where candidates lose it

What the interviewer asks next

Try it first

Where does the update rule come from?

How do you get the iterates without a calculator?

Why is it the digits that double, and when does that fail?

Where candidates lose it

What the interviewer asks next

Try it first

Which numbers share a factor with 1,000?

Why does the quick product formula work here?

Where does a question like this lead in an interview?

Where candidates lose it

What the interviewer asks next