Take on the challenge

We love problems at Sparx.
Every day we’re tackling brain-aching and complex issues as we continually explore new methods and ideas. We’ve created the set of problems below to give an idea of the kind of thinking that goes on at Sparx. And yes, because it’s fun and the kind of thing we like to do. They’re sorted and colour coded by difficulty level – see how tough you can go!

Enjoy these? Then you’d probably fit right in at Sparx and we’d love to hear from you. Get in touch here.

  • You're on a TV gameshow

    on the verge of winning a luxury round-the-world cruise. In front of you is a giant spinner bearing the numbers 1 to 10 (with each number being equally likely to come up). To win, all you need to do is spin the wheel three times and beat your previous spin each time. What is your probability of winning the cruise?

    Answer
  • You're Blindfolded

    and a deck of 52 playing cards is placed in front of you. You're told that the 13 hearts are face-up, and all the others are face-down, but the ordering of the cards is completely random. Without peeking, how can you rearrange the deck into two piles such that each pile has the same number of cards facing up?

    Answer
  • Today's your Birthday

    and it's time to cut the cake, which measures 12 inches wide by 12 inches deep by 6 inches tall. As usual, there is icing on the cake's top and sides, but not on the bottom. How can you cut the cake up into 9 pieces so that every piece has the same amount of cake and the same amount of icing?

    Answer
  • The problem cards in this deck are colour-coded by difficulty level, as follows:

    Easy


    Find two whole numbers that multiply together to give one million, with the added requirement that neither number may contain the digit zero.

    Jim builds a tower by stacking three dice on top of one another (each one being a standard six-sided dice). If the number showing on the top face of the tower is 6, what is the sum of the numbers on the five hidden faces?

    If it takes three workmen three days to dig three wells, how many days would it take six workmen to dig four wells?

    The number 64 has the unusual property of being both a square (8 x 8 = 64) and a cube (4 x 4 x 4 = 64). What is the next highest number with this property?

    Roxy’s sock drawer contains a jumble of 16 socks; 4 red, 4 yellow, 4 green, and 4 blue. If she reaches into the drawer and grabs two socks at random without looking, what is the probability that the socks she picks are the same colour?

    Moderate


    Pick any three-digit number (e.g. 852) and write it down twice in a row to create a six-digit number (e.g. 852852). You'll find that this number is always exactly divisible by 143 – for instance, 852852 = 143 x 5964. Can you explain why?

    If you roll 10 standard six-sided dice at the same time, what is the most likely total score?

    The "triangular numbers" (1, 3, 6, 10, 15, 21, 28, 36, 45, ...) can be visualised as the number of balls in a steadily growing triangle (since 3 = 1+2, 6 = 1+2+3, 10 = 1+2+3+4, 15 = 1+2+3+4+5, and so on). What is the first triangular number to exceed half a million?

    Suppose you are blindfolded, and a deck of 52 playing cards is placed on the table in front of you. You are told that the 13 hearts in the deck are face-up, and the remaining 39 cards are face-down, but the ordering of the cards is completely random. Without peeking, how can you rearrange the deck into two piles such that each pile has the same number of cards facing up?

    Farmer Giles owns a large flat field with a barn at the centre. The barn is a perfect cube that measures 8 metres along each side. During the summer months, Farmer Giles attaches his goat Billy to one of the outside corners of his barn (at ground level) with a 12-metre long rope. What is the area of the region that Billy able to reach? (Note: To get an answer within 1% of the true value using just pencil and paper, you could approximate π as being roughly 5% more than 3 – i.e. 3.15).

    Tough


    The final stage of the 2013 World Rock-Paper-Scissors Championship was a round-robin tournament in which each pair of players met exactly once. In each head-to-head match, 2 points were awarded for a win, 1 for a draw, and 0 for a loss. If the top four scores in the tournament were 9, 8, 5, and 3, what was the lowest score, and how many players took part?

    Three people stand before you: two Scientists (who will always answer questions truthfully) and a Scoundrel (who may answer either truthfully or falsely, depending on their mood at the time). All three of them know who's who, but you haven't the faintest idea. How many questions do you need to ask to identify a Scientist with 100% certainty?

    Princess Naturalia has chosen three natural numbers - X, Y, and Z - and to be deemed worthy of entering her castle, you must determine her three numbers by asking just two questions. However, you're told that the princess will only answer a very specific type of question - namely, she'll tell you the value of AX + BY + CZ for any natural numbers A, B, C of your choice. Naturally, your second question can be conditional on her answer to the first question. What should your strategy be? (Note: A natural number is a positive integer - i.e. a whole number greater than zero.)

    Lennon the beetle lives on a hexagon made from six large beams of wood. One morning, he decides to go for a random stroll on the hexagon, starting from his home at corner H. At every stage of the walk, Lennon randomly picks one of the two edges adjacent to his current corner (each being chosen with probability 1/2) and crawls along the edge to the next corner. There are no restrictions on retracing the edges - for instance, it is possible that Lennon could arrive back at corner H after two moves. What is the probability that Lennon stands back at corner H after exactly six moves?

    The final round of a TV gameshow features a giant spinner bearing the numbers 1 to 50. When the wheel is spun, each of the 50 numbers is equally likely to arise. The contestant spins the wheel three times, and to win the star prize, the number displayed on the wheel must increase each time (i.e. the contestant's second spin must exceed the first, and the third must exceed the second). What is the probability of the contestant winning the star prize?

    Fiendish


    Bob is browsing through some books at the library when he finds a stack of pages that someone has ripped out of another book. If the page numbers in the stack are consecutive and add up to 808, can you work out which page numbers were torn out?

    Grace has written down a list of positive integers (not necessarily all different) that sum to 21. What is the highest possible value for the product of her numbers? (Note: The product is the value obtained by multiplying all the numbers together.) Does your answer change if the numbers do not have to be integers?

    To play the game of "Go Nuts", all you need is six bowls, containing 1, 4, 9, 16, 25, and 36 nuts respectively, and a large bag of spare nuts. Two types of move are permissible: (1) transferring any number of nuts from one bowl to one of the other bowls, or (2) adding seven more nuts to the collection - either all in the same bowl, or split across two different bowls. The goal is to equalise the number of nuts in each of the six bowls. Is it possible to achieve this? If so, how many moves are required?

    When you studied probability at school, do you remember drawing a 6x6 grid to represent the possible outcomes of rolling two standard six-sided dice - and figuring out that the most likely total was 7, with a probability of 1/6? Ah, happy days. Here's a rather more absorbing question, though. Can you design two non-standard six-sided dice (by changing the number of spots on each face) so that when the dice are rolled together, the probability of getting each total from 2 to 12 is exactly the same as for two standard dice? The dice you design may be different from one another, but every face must have at least one spot (no blanks are allowed).

    A basketball player decides to spend an afternoon practicing free throws and recording her performance. Over the first hour, her scoring percentage is less than 75%, but by the end of the session, her overall scoring percentage for the day is more than 75%. Is it necessarily true that at some moment during the session her scoring percentage was exactly 75%? Now let's reverse the situation, i.e. her scoring percentage is more than 75% over the first hour, but her overall scoring percentage at the end of the day is less than 75%. Is it necessarily true that at some moment during the session her scoring percentage was exactly 75%?