BEBRAS 2019 Grade 7-8 Round 2 Ext

The beavers in Bobby's class want to play football in PE next week. Bobby allows it only if all three conditions hold: it is sunny; humidity is low or normal; no other class is using the pitch. The forecasts and bookings are shown below.





Monday: sunny, low humidity, Ms Garcia booked. Tuesday: rainy, high humidity, Mr Smith booked. Wednesday: rainy, normal humidity, unbooked. Thursday: sunny, low humidity, unbooked. Friday: sunny, very high humidity, Ms Parker booked. On which day can they play?

Lucilla is learning to use a fork. Her mother says outside pizza pieces, which have crust, may be eaten by hand; inside pieces, which have no crust, must be eaten with a fork. How many inside pieces are shown?

The mock battle is one of the most eagerly awaited activities at the Bebras festival. Each beaver brings a pillow as their “weapon”. Beavers with the same pillow pattern form a team. Every team has exactly 3 members. If more than 3 bring a pattern, only 3 may play; if fewer than 3 bring it, none may play. The pillows are shown below. How many beavers cannot play?

A beaver lives in a small pond surrounded by forests and rocky regions. Each day it cuts down every forest region adjacent to the current pond and turns those regions into pond. Rock cannot be converted.

Pond symbol:



Forest symbol:



Rock symbol:





After how many days will all forests on the map have been cut? Editorial adjacency convention: regions are adjacent when they share a boundary segment; new pond regions expand further only on the next day.

Mawar plays ring toss, trying to throw each ring around the peg.



The first through fifth throws score 5,4,3,2,1 respectively if successful, and 0 if unsuccessful.



How many points does Mawar score for the final arrangement below?



Editorial stacking convention: rings keep their landing positions and a later ring lies above an earlier ring where they overlap; a throw succeeds exactly when the peg lies inside its ring.

Bob earns 40 coins per hour and starts work at 8:00. Each completed 15 minutes of lateness costs one quarter of the hourly wage. Arriving at 8:11 incurs no fine; arriving at 8:20 incurs one 15-minute fine. Train services are:



Slow: from 6:00 every 5 minutes,40-minute journey, fare 5. Regular: from 6:00 every 10 minutes,30-minute journey, fare 10. Fast: from 7:00 every 15 minutes,20-minute journey, fare 15. Express: from 8:00 every 30 minutes,10-minute journey, fare 20. Bob reaches the station at 8:08 after oversleeping. Which train minimizes fare plus late fine?

Ali has been captured and sends detective Lim “5 bodyguards are watching us” using the 6 × 6 letter-digit table. Rows and columns are numbered 1–6.



Rows are ABCDEF; GHIJKL; MNOPQR; STUVWX; YZ 0123;456789. The example encoding is 62+12331451214311361441+113615+4511421322233221+4341. Lim replies “2 Agent Beaver will come”. Which is its encoding?

Ben, the log-market accountant, is ill. Since the beavers do not trust anyone else to replace him, he tells them to use this recording system: buyer and seller publicly announce every trade; everyone records [transaction ID,buyer ID,seller ID,number of logs sold]. At day's end, a trade is valid if at least half the records are correct; otherwise it and affected later trades are invalid. For each trade the buyer pays a total fee of 10% of its value to the beavers who recorded it correctly.

Eight beavers A–H each start with 100 logs. The records and correct recorders are:



01: F buys 80 from B; correct A,B,C,D,E,F,G,H.
02: C buys 60 from G; correct A,B,C,D,G,H.
03: A buys 90 from F; correct A,F,G.
04: B buys 100 from F; correct A,B,C,D,G.
05: H buys 140 from A; correct A,B,C,D,E,G,H.
Who owns the most logs at the end?

Editorial accounting convention: fees are paid in logs, and an invalid trade transfers neither logs nor fees; a seller cannot sell logs they do not own. No equal division of the fee is needed to determine the answer.

Alice and Bob alternately remove marbles from a bag of 28. On a turn they must remove a positive square number 1,4,9,… that does not exceed the number remaining. Whoever takes the last marble wins. Alice starts and both play perfectly. How many should Alice take on her first move to force a win?

Angel,Brian,Cindy and Danny each receive five cards numbered 3,4,5,6,7, in the orders below.



Player 1:6,5,7,3,4. Player 2:3,5,4,6,7. Player 3:7,6,5,4,3. Player 4:4,5,6,3,7. Each round everyone plays the next card. The highest card wins 2 points; if several tie for highest, each receives 1 point. Repeat for all five cards. Brian has the highest total. Which player is Brian?

A pirate beaver has a huge square treasure map divided into small squares. The boat is too small to carry it, so the beaver cleverly compresses it into a tree in a notebook. If every small square in a region has the same color, draw a square of that color. Otherwise draw a circle, divide the region into four equal quadrants, and recursively encode them. The examples establish child order: upper-right 1, upper-left 2, lower-left 3, lower-right 4.



Which of the four maps is encoded by this tree?

Xavier,Ylenia and Zoe sometimes play the lottery. They want to learn whether someone won while keeping the winner's identity secret. Each pair meets and tosses a coin together. Each person announces whether their two observed tosses were the same or different. Everyone tells the truth except the winner, who lies. The example shows Zoe winning and all three announcing “different”.



In the example the Zoe–Xavier toss is tails, Xavier–Ylenia is heads, and Zoe–Ylenia is tails. Now Xavier announces “same”, Ylenia “same”, Zoe “different”, and at most one person won. Who won?

A shop sells 10 candy types priced 1,2,…,10 coins. Smart buys 3 pink spiral lollipops,4 green lollipops and 2 yellow wrapped candies for 47 coins.







Smart has only 46 coins, so exchanges one green lollipop for one pink spiral lollipop and the price becomes exactly 46.





How much does one yellow wrapped candy cost?

A wizard has a box that combines two jewels into one. The input jewels must have different shapes, and both their shapes and colors must follow a listed rule.



Shape rules: triangle+square→circle; triangle+pentagon→square; triangle+hexagon→pentagon; square+pentagon→hexagon; square+hexagon→star; pentagon+hexagon→triangle.



Color rules: B+Y=R; B+R=G; B+G=O; Y+R=B; Y+G=P; P+O=Y.

The initial jewels are a Y triangle, two G pentagons and a B square.



Which jewel cannot be obtained?



Editorial rule convention: inputs are consumed, operations may be repeated, and only combinations explicitly listed in both rule tables are allowed.

Each cell of a 3 × 5 grid either contains a dot or is empty. The cells must not all contain dots. Every contiguous 2 × 2 square must contain exactly 1 or 3 dots. An example is shown. How many such 3 × 5 grids are there?