Question bank BEBRAS 2018

Benjamin fills each 3 × 3 grid with pictures so no row or column contains a repeated picture. He made a mistake in one grid. Which one?

Peter helps his mother bake a cake. Her instructions are:
1. Add butter to the bowl.
2. Add sugar.
3. Repeat three times: add one egg, then mix the bowl thoroughly.

Which procedure follows these instructions?

Two sizes of dogs stand in this order.



One swap exchanges two neighbouring dogs. What is the minimum number of swaps needed to make the three large dogs stand together?

A car park has 12 spaces. These pictures show the occupied spaces on Monday and Tuesday.



How many spaces were unused on both days?

Honomakato has islands Ho, No, Ma, Ka, To. A large cable joins the mainland to Ho. Smaller cables join Ho–No, Ho–Ka, Ka–Ma and Ka–To.



Which extra cables ensure every island keeps Internet access even if any one small cable breaks?

The friendly beavers of Jeju leave messages for visiting neighbours using a special three-bar gate.



Four example messages are: I am home; I will return late; I will return soon; nobody is home.



Cheolsoo thinks more messages are possible by changing the bars. What is the largest number of messages?

Editorial reading of the examples: each of the three distinct bar positions can independently be across the gate or lowered at its free end.

John wants to visit nine cities: Onecity, Twoland, Threecity, Fourland, Fivecity, Sixland, Sevencity, Eightland, Ninecity. Bus routes connect Threecity–Onecity, Ninecity–Twoland, Threecity–Ninecity, Ninecity–Onecity, Onecity–Twoland, Eightland–Ninecity, Sixland–Sevencity, Fivecity–Fourland and Fourland–Eightland. Starting at Threecity, which pair consists of cities he cannot visit?

Zac starts at 1 and walks to 2; Pan starts at 2 and walks to 1. Letters B, E, R, A, S stand beside the path, but each beaver reads only letters on their left while walking.



Which pair of letter sequences do they read?

Seven skaters start in the pictured order.



After every minute, the front skater moves to the back to save energy. Who is at the front after one hour?

A smile detector first replaces a face by two eye dots and a mouth line.



It places this sketch over a test template with two red eye circles, a red mouth boundary and four black contact points.



It accepts a smile only when the sketch touches the black points without touching the red outlines. The examples show an accepted face and a rejected face.



How many of these ten sketches are accepted?

Count binary strings (made only of 0 and 1) with no two adjacent ones, such as 010010001. The table records counts for length N.



For N=1 the strings are 0,1; for N=2 they are 00,01,10; for N=3 they are 000,100,010,001,101. How should the complete count row read?

Martin travels from home H to work W by metro and must change trains. The map labels stations with circles and gives the time between adjacent stations.



Assume changing trains takes no time. What is the shortest journey time?

Ivan builds a stepped pyramid from identical cubes. Its layers, from bottom upward, contain 25,16,9,4,1 cubes as shown. He paints every outside face, including the bottom.



How many small-cube faces does he paint?

Two roads connect Beavertown and Treeville. Road 1 has a bridge allowing weight at most 7 tonnes. Road 2 has a tunnel allowing height less than 3 m and a ferry allowing length less than 7 m.



Vehicle labels give height, weight, length in that order: for example 180 cm, 6 t, 8 m means height 180 cm, weight 6 tonnes, length 8 m. Which vehicle can use either road?

The park has six letter-labelled trees. A letter is encoded by L and R turns from the entrance arrow, taking left or right at each junction.



For example A is LLL and R is LRR. Concatenating codes makes words: RLLLLRRLLL encodes SARA. How many L/R characters encode BEBRAS?

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?