Every card in Bebras thinking toolkit

11 times

HintWork out the net gain from one whole run of the block, then count how many of those gains fit the total.

WhyOne run of the block moves the robot forward 5 and back 2, a net gain of 3 squares — so tracing every single move is wasted effort. The robot has gone from square 1 to square 34, a gain of 33, and 33 ÷ 3 = 11 runs. Written as algebra that is 1 + 3n = 34, the equation a Year 9 meets in an algebra lesson. The landing squares are 1, 4, 7, 10 …, always one more than a multiple of 3, so a follow-up asking whether the robot can finish on square 30 has the answer no, however long the loop runs.

An algorithm

HintNamed after al-Khwarizmi, a ninth-century Persian mathematician.

WhyA recipe, a knitting pattern and a set of directions are all algorithms: each is written so that a follower with no imagination still gets the right result. That is exactly why a computer can run one. Bebras puzzles hand you an algorithm and ask what it produces — or hand you a result and ask which algorithm made it.

sequence; selection; repetition

HintEvery program ever written uses just these three; each name is a noun ending -ence or -tion.

WhyEvery program ever written is made from just these three, which is why Bebras keeps asking about them. Sequence is the default: one step, then the next. Selection is a fork with a test — IF the tile is black, turn left, ELSE carry on. Repetition is a loop: "repeat 4 times" or "until you reach the wall". Spotting which one a puzzle is really about tells you how to trace it.

double

HintThink what happens to a place value each step left in decimal — then shrink the multiplier.

WhyBinary is base 2: the place values are 1, 2, 4, 8, 16, 32 … the powers of 2, where ordinary numbers use 1, 10, 100, 1000. A digit can only be 0 or 1, so a binary number is really a row of switches that are on or off, and its value is the sum of the place values that are on. Computers store everything this way, and Bebras puzzles about light bulbs, flags or coins in a row are nearly always binary in disguise.

16

HintEach digit has two choices, and the choices multiply.

WhyEach bit is 0 or 1, so four bits give 2 × 2 × 2 × 2 = 16 patterns, from 0000 to 1111. Every extra bit doubles the count: eight bits — a byte — give 256, which is why so many Bebras answers land on a power of 2.

Decomposition

HintThe same word describes a fallen leaf breaking down into simpler parts.

WhyDecomposition is one of the four computational-thinking habits, alongside pattern spotting, abstraction (ignoring the details that don't matter) and algorithm design. In a puzzle it means: sort out bus 1 completely before you think about bus 2; deal with the seating rule before the luggage rule. Small problems have small answers you can check, and the big answer is just those put together.

7

HintHalve the range with every question and count how many halvings take 100 down to a single number.

WhyAsk "bigger than 50?" and half the numbers vanish whatever the answer: 100 → 50 → 25 → 13 → 7 → 4 → 2 → 1 is seven halvings. Because 2⁶ = 64 is less than 100 but 2⁷ = 128 is more, six questions can't always do it and seven always can. This is binary search — the reason a sorted list is so quick to search, and why Bebras loves things in order.

A node

HintAlso the word for the point on a plant stem where a leaf grows.

WhyA node is also called a vertex; the lines joining nodes are called edges, and the whole picture is a graph — not the x–y kind with axes, the dots-and-lines kind. Computer scientists draw towns-and-roads, people-and-friendships and tasks-and-what-must-come-first all as the same picture, so one method works for all of them. A Bebras puzzle about who can pass a message to whom is really a puzzle about a graph.

Add up each route's lengths and pick the smallest total

HintFewest roads does not mean fewest kilometres.

WhyWrite the running total at each town as you go, and if you reach a town again by a second way with a smaller total, cross the old one out. That is the idea behind Dijkstra's algorithm, the method sat-navs use. Counting roads instead of adding lengths is the classic trap — one long motorway can be worse than three short lanes.

Upside down

HintTwo flips cancel out, so only the spare one counts.

WhyOdd or even is the whole story: every pair of flips cancels, so the cup's position depends only on whether the number of flips is odd or even. That is an invariant — something a move can never change, or changes in a fixed way. Bebras puzzles about switching lights, swapping cards or hopping between squares on a chessboard are often settled in one line by parity, however many moves are involved.

6

HintFix the first letter and count the ways the other two can go.

WhyListing in a fixed order — every code starting with A, then B, then C — is what guarantees you have found each one and none twice: ABC, ACB, BAC, BCA, CAB, CBA. Each of the three starting letters leaves two ways to arrange the rest, so 3 × 2 × 1 = 6. Bebras counting puzzles are won by an organised list, not a hopeful guess, and the rule you can trust grows out of the small list you made.

The top 10%

HintIt is the rarest certificate of the lot, and it is judged nationally rather than within your own school.

WhyBebras is run in the UK by the Raspberry Pi Foundation: a free, 45-minute online challenge sat in school, with Intermediates the category for ages 12–14 (Years 8–9) and the 2026 window running 9–20 November. Gold goes to about the top 10% of each category nationally; within your own school, Distinction goes to the top 25% and Merit to the next 25%. No coding is needed — the puzzles test logic, algorithms, patterns and data. Only schools that took part can then enter their students for the follow-on Raspberry Pi Foundation Coding Challenge in the spring, where the coding starts.