BEBRAS 2018 Grade 7-8 Round 2 Ext

A beaver wishes to become a ninja and chooses the nickname Zukame Moru using the alphabet below. What is the beaver's real name?



The complete mapping is A–ka, B–zu, C–mi, D–te, E–ku, F–lu, G–ji, H–ri, I–ki, J–zu, K–me, L–ta, M–rin, N–to, O–mo, P–mor, Q–ke, R–shi, S–ari, T–chi, U–do, V–ru, W–mei, X–na, Y–fu, Z–zi.

Lara defines the nonuniformity of a string as the largest frequency of a character minus the smallest frequency among characters that occur. For abacaba it is 4−1=3. Changing two characters can give abbccba, whose nonuniformity is 1. Lara has ababaaaaaaccabc and may replace 6 characters with letters of the alphabet. What is the smallest possible nonuniformity?



Source wording note: the final source sentence says “uniformity”, while the definition and objective concern nonuniformity; use the defined largest-minus-smallest quantity. The illustration circles 2 beside abacaba, but its frequencies 4,2,1 actually give 3; the original graphic is retained.

A fashion shop makes bracelets from the two pairs of ornaments shown. It starts with one pair, then inserts further complete pairs between the two ornaments of any existing pair, as in the examples. Which bracelet follows this rule?

A car park has 12 spaces. The diagrams show occupied spaces on Monday (left) and Tuesday (right). How many spaces were unused on both days?

Zac starts at 1 and walks to 2. Pan starts at 2 and walks to 1. The letters B,E,R,A,S are placed along the road, but each beaver can read only signs on their left. What sequences do they read?

A repeated DNA string has even length and identical left and right halves. Alex looks for the longest contiguous repeated DNA substring in Vlad's DNA AACGTACGTACCTTACTAC. What is its length?

Bobi needs seven 4 m logs and seven 3 m logs to build a dam. Only 10 m logs are available. What is the smallest number of 10 m logs that must be cut? Ignore the thickness of cuts.

Ten beavers need computers to write and edit their school newspaper. Red cells in the schedule mark the time slots each beaver needs. All computers are identical and at most one beaver can use a computer at a time. What is the minimum number required?



Columns run 8:00,9:00,10:00,11:00,12:00,13:00,14:00; rows are beavers 1–10.

NUMO sorts a row of numbers. In each pass, it activates buttons from START to END. A button compares the two numbers immediately below it: if the left number is at most the right, it does nothing; otherwise it swaps them, then proceeds. Passes continue until the row is increasing from left to right.





Bubble says that each new pass can stop one button earlier than the previous pass, making sorting faster. Which statement is correct?

Antonio plants olives in an N-by-N square and surrounds them with cypress trees to protect them from wind, following the patterns N=1,2,3. How many olive and cypress trees together are needed for N=5?

Nine members of Bebragram have the initial profile colors shown. An edge means friendship. Everyone wants a white profile. At 8 am each day a member may turn white if at least half of their friends are white. What is the minimum number of days until all nine are white?



Editorial timing convention: updates at the same 8 am are simultaneous, based on the colors just before that update; white profiles stay white.

Kim must walk from home (Nhà) to the grocery shop (Cửa hàng), avoiding all rocks and trees. Each step goes one cell left, right, up or down. A friend is visiting, so Kim chooses a shortest route. How many shortest routes are there?

A beaver family builds four rooms, five connecting passages and seven outside doors as shown. The young beavers find a walk that uses every passage and every outside door exactly once. In which room must they start?

A vacuum robot has commands F (move forward one cell,1 minute), R (turn right 90°, no time), and W (clean the current cell,1 minute). It cannot enter a cell occupied by furniture. It starts and finishes at corners chosen from A,B,C,D of this 6 × 6 floor. What is the minimum time to clean every unoccupied cell?



Editorial clarification: the starting and finishing corners may differ; every free cell, including corner cells, requires one W operation. Revisited cells need not be cleaned twice.

Moff sends Manu a message using logs joined by ropes. Each log has one,two or three neighbors. Words appear only on logs with one neighbor. Exactly one log has two neighbors; using it as the root allows the tree to be arranged as shown. The example reads “I am a beautiful beaver” from left to right in its meaningful arrangement.



For the letter tree below, which message cannot be obtained?