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  1. 002A 10 x 10 x 10 cube is built from small unit cubes and the outside surface is painted green. How many small cubes have green paint on them, and how many have none?Mental maths and countingCoreTD SecuritiesNew York · 2024

    Try it first

    Quick instinct: how many small cubes are never painted?

    Show the worked solution

    488 small cubes are painted and 512 are not. The unpainted ones form a hidden core with one layer peeled off every side, so the core is 8 x 8 x 8, which is 512. The whole cube is 1,000, so 1,000 minus 512 leaves 488 with at least one painted face.

    Why count the cubes you cannot see?

    Picture a mango with its skin peeled. Measuring the skin piece by piece is fiddly; measuring the fruit left inside and subtracting from the whole is easy. The unpainted cubes form one clean block, so counting them and subtracting is faster and harder to get wrong than adding up the painted surface. Peeling one layer from both sides of each dimension takes 10 down to 8.

    One face of the 10 x 10 x 10 cube, by painted faces per small cube8 x 8 innerCorners, 3 painted faces8 corners8Edges, 2 painted faces12 edges x 896Face centres, 1 painted face6 faces x 64384Painted at least once488Never painted: the 8 x 8 x 8 core512
    Each face of the big cube shows 8 corner cubes with three painted faces, 96 edge cubes with two and 384 face-centre cubes with one. They add to 488 painted cubes, and the 8 x 8 x 8 core of 512 cubes is never painted.

    How do you check 488 the other way?

    Count the surface directly, taking care not to double count. There are 8 corners, 12 edges each carrying 8 cubes once the corners are removed, and 6 faces each carrying an 8 x 8 centre. That gives 8 plus 96 plus 384, which is 488. Two methods agreeing is exactly what the interviewer wants to hear.

    The relationship
    painted=n3−(n−2)3=1000−512=488\text{painted} = n^3 - (n-2)^3 = 1000 - 512 = 488
    ncubes along one edge, here 10
    (n-2)^3the hidden core after peeling one layer from each side
    What it says in wordsThe painted cubes are everything minus the core you cannot see.

    Where candidates lose it

    The fast wrong answer is 600: six faces of 100. It counts every edge cube twice and every corner cube three times. Candidates who start from the surface usually end up correcting on the fly and lose the room.

    Start from the inside. Say the core formula, give 512 and 488, then offer the corner, edge and face split as a check.

    What the interviewer asks next

    • How many small cubes have exactly two painted faces?
    • What size cube has as many unpainted as painted small cubes, roughly?
    • Now paint only the top and bottom. How many cubes are painted?

    Asked at TD Securities, Generalist, New York, 2024 (Wall Street Oasis): If I gave you a 10x10x10 cube and I painted the surface green, how many cubes are green?

  2. 015Without a calculator, what is 9 to the power of 6?Mental maths and countingCoreNomuraTokyo · 2025

    Try it first

    Before computing anything, which size is right?

    Show the worked solution

    531,441. Rewrite 9 to the 6th as the square of 9 cubed. 9 cubed is 729. Then square 729 by splitting it into 700 plus 29: 700 squared is 490,000, twice 700 times 29 is 40,600, and 29 squared is 841. They add to 531,441. As a check, it should be a little over half a million, because 0.9 to the 6th is about 0.53.

    How do you keep every step in your head?

    Think of carrying a heavy box up six floors: you take it in two flights of three with a rest between, not six lifts in a row. Rewriting a power as the square of a smaller power cuts the work to one easy cube and one square you can split. 9 squared is 81, and 81 times 9 is 729, so 9 cubed is 729. Then 9 to the 6th is 729 squared, and squaring a number near a round one is the most practised move in mental maths.

    Square the cube: 729 squared, split as 700 + 299 x 98181 x 9729 = 9 cubed9 to the 6th= 729 squared700 x 700490,000700 x 29 = 20,300700 x 29 = 20,3008417002970029Pieces not drawn to scaleAdd the pieces490,000+ 20,300+ 20,300+ 841531,441Checks: ends in 1;digits sum to 18
    Nine cubed is 729, and squaring 729 as 700 plus 29 gives a 490,000 block, two strips of 20,300 and an 841 corner, which add to 531,441.
    The relationship
    96=(93)2=7292=(700+29)2=490,000+2(700)(29)+292=531,4419^6 = (9^3)^2 = 729^2 = (700+29)^2 = 490{,}000 + 2(700)(29) + 29^2 = 531{,}441
    9^3729, nine cubed
    (700 + 29)^2the square split into a round part and a remainder
    2(700)(29)the two strips, 20,300 each and 40,600 together
    What it says in wordsSquare a number by splitting it into a round part and a remainder; the cross term counts twice.

    How do you check it before saying it?

    Three quick checks. An even power of 9 ends in 1, because 81 does, and 531,441 ends in 1. Any power of 9 is divisible by 9, so its digits must sum to a multiple of 9: 5 + 3 + 1 + 4 + 4 + 1 is 18. And the size is right: 9 is 10% below 10, and six 10% trims leave about 53% of a million. A digit check and a size check together catch nearly every slip in the cross term, which is where most mental errors happen.

    A second route, if you prefer it: 9 to the 6th is also 81 cubed. 81 times 81 is 6,561, and 6,561 times 81 is 6,561 times 80 plus 6,561, which is 524,880 plus 6,561, giving 531,441 again. Saying one route and confirming with the other is worth more than speed.

    Where candidates lose it

    The usual loss is multiplying 9 by itself six times in a row and dropping a digit somewhere in the five-digit middle steps. By the time you reach 59,049 times 9, the interviewer has watched you struggle for a minute.

    The other loss is saying the number without a check. Give the answer, then the two-second test: it ends in 1 and its digits sum to 18.

    What the interviewer asks next

    • What is 11 to the power of 4?
    • What are 99 squared and 999 squared?
    • Is 2 to the power of 20 bigger or smaller than a million, and by how much?

    Asked at Nomura, Generalist, Tokyo, 2025 (Wall Street Oasis): I think one of the math questions was 9 to the power of 6