Cambridge A Level Thinking Skills 9694 — 2014 May/June Paper 3 · Variant 2

This document consists of 7 printed pages, 1 blank page and 1 insert. IB14 06_9694_32/3RP © UCLES 2014 [Turn over *2946340790* Cambridge International Examinations Cambridge International Advanced Level THINKING SKILLS 9694/32 Paper 3 Problem Analysis and Solution May/June 2014 1 hour 30 minutes Additional Materials: Electronic Calculator READ THESE INSTRUCTIONS FIRST An answer booklet is provided inside this question paper. You should follow the instructions on the front cover of the answer booklet. If you need additional answer paper ask the invigilator for a continuation booklet. Answer all the questions. Calculators should be used where appropriate. The number of marks is given in brackets [ ] at the end of each question or part question.

2 © UCLES 2014 9694/32/M/J/14 1 Study the information below and answer the questions. Show your working. The following diagram shows the design for a toy. It represents a solid box with a single entry point (shown with an arrow) through which a ball is inserted into the maze. Inside the box there are five straight, intersecting passageways, represented by white areas in the diagram. To create the maze a number of bungs, represented as black squares in the diagram, are placed inside the passageways, ensuring that no part of the maze is blocked off (i.e. no white space is unreachable by the ball). In the diagram, bungs have been placed at the squares labelled A4, B3 and D5. A B C D E F G 1 2 3 4 5 N S E W (a) When creating a new maze, with no bungs currently in position, in how many different places can the first bung be put without blocking off any section of the maze space? [2] (b) List all the pairs of places where just two bungs would block off a section of the maze, even though neither bung on its own would. [2] (c) A white square is called a ‘pocket’ if it is open on only one side (e.g. B1 in the diagram above). What is the greatest and least number of pockets that a maze could have, if it contains three bungs, and no areas are blocked off? [2] Once the bungs have been placed, the ball is inserted. The box is shaken around, and the player is given the task of trying to get the ball out, by tilting the box and thus making the ball roll in different directions. The player must wait for the ball to stop before changing the direction of tilt. The maze can be tilted in 8 directions, which can be referred to using the standard compass directions. When the maze is tilted in one of the four principal directions (N, E, S, W), the ball moves in that direction until it reaches an obstruction, where it stops. If it is tilted in one of the other four directions (NE, SE, SW, NW), the ball moves diagonally where possible, but if not it moves at 45° to the direction of tilt. An example of the effect of a south–west tilt is shown in the diagram below. A B C D E F G 1 2 3 4 5 N S E W (d) Consider the maze represented at the beginning of the question, with bungs at A4, B3 and D5. State the starting square from which the ball cannot escape with fewer than three tilts. List three directions, in the order that they occur, that will allow the ball to escape. [2] (e) State where two bungs should be placed, and where the ball should start, if it is to require a route involving four tilts in order to get out. [2]

3 © UCLES 2014 9694/32/M/J/14 [Turn over 2 Study the information below and answer the questions. Show your working. A door has a lock which is operated by entering a four-letter code into an electronic keypad. There are 4 buttons on the keypad, labelled A, B, C and D. The program in the keypad operates by moving between five different states. Every time that a button on the keypad is pressed the program either moves to a different state or remains in its current state. When it moves into state 5 the door is released and the program moves back to state 1. The table below shows the next state for the program when different inputs are made while it is in states 1 to 4. Input A B C D State 1 1 1 2 1 2 3 1 2 1 3 1 1 4 1 4 3 1 2 5 The correct first character of the code is C (which moves the program from state 1 to state 2). Mark attempted to unlock the door by entering the sequence C B C A. When the C button was pressed for the first time the program moved to state 2. (a) Show that Mark has left the program in state 3. Write down which state the program was in after each time Mark pressed a button. [2] (b) What two further button presses would unlock the door? [1] If the program is in state 2, then pressing either B or D will return the program to state 1, as neither is the correct second character of the code. C is also incorrect, but returns the program to state 2 because C is the correct first character of the code. (c) Explain why the program is designed so that it will always be in state 1 after B is pressed. [1] (d) Explain why the program is designed so that pressing A while the program is in state 4 will return the program to state 3, rather than state 1. [2] Another lock is programmed to open only when the code AABDAC is entered. The program for this lock uses seven states. When it moves into state 7 the door unlocks and the program immediately moves back to state 1. (e) Draw a table to show the next state for the program when different inputs are made while it is in states 1 to 6. [4]

4 © UCLES 2014 9694/32/M/J/14 3 Study the information below and answer the questions. Show your working. Tossing a coin is not a good way to make a fair random selection of one of three people, but a fair choice can be made by rolling a standard die. (a) Suggest a simple way to choose fairly between three people, using a single roll of a die. [1] Alexander noticed that a single roll of a die is sufficient to select fairly from 12 possible options. Since there are 12 edges, he could do this by looking at the vertical edge which is nearest to him. Each of the 12 options could be associated with one of the 12 edges. Charlotte was worried that sometimes the die would land with a face ‘square on’. They decided that in such cases they would always take the edge to the right, as this would not introduce any bias. Zoe pointed out that she could do the same much more easily by rolling the die, then considering both the number on top and whether the vertical edge nearest her was on the right or left of centre. Each of the 12 options could be associated with a number and a direction (1L, 1R, 2L, 2R etc.). Alexander then worked out how to choose one from 24 options fairly using a single roll. (b) Describe a method he could use. (Do not list all the cases.) [2] Charlotte suggested subtracting the number on the left visible vertical face from the one on the right. In the case shown above that would give 4 – 6 = –2. (Note that opposite faces of a die add to 7.) (c) (i) How many different results are possible? [2] This method does not give the same chance of each result. (ii) How many times more likely is a difference of 2 than a difference of 3? [2] Donald obtained a regular twelve-faced solid (dodecahedron) with faces numbered 1 to 12, and on which opposite faces always add to the same sum. (d) What is the sum of opposite faces? [1] Donald could use the 12 faces to select one of 2 options, or one of 3, 4, 6, or 12 options. John was much more interested in using a dodecahedron to choose one of 5 players to start a game. (e) How can John achieve this in an unbiased way using a single roll? [3]

5 © UCLES 2014 9694/32/M/J/14 [Turn over (f) (i) Suggest a way for Alexander to get 60 equally likely options from one roll of a dodecahedron. [1] (ii) Zoe finds a way to get 120. How can she do this? [1] (g) Charlotte says that by rolling the standard die and the dodecahedron once each, she can fairly choose between 45 options. Suggest a way that she could achieve this. [2]

6 © UCLES 2014 9694/32/M/J/14 4 Study the information below and answer the questions. Show your working. Trypples is a game for two players, played over a number of rounds. It involves the placing of tiles onto a 3 x 3 grid of squares. There are 12 tiles, as follows: 1 1 2 2 3 3 4 4 5 5 6 6 Before the game, the toss of a coin decides who will be player X and who will be player Y. In each round, the players take turns to remove one tile, at random, from a bag and place it onto one of the empty squares on the grid, until the grid is complete. The player placing the first tile alternates from round to round, with player X going first in round one. The object of the game is to create lines of three different numbers. Each such line scores the sum of the three numbers. Lines in which a number appears twice score zero. Throughout the whole game, player X scores points for all (horizontal) rows and player Y scores points for all (vertical) columns. For example, in the completed grid shown below, player X scores 17 points (1 + 2 + 6 and 1 + 5 + 2) and player Y scores 11 points (6 + 3 + 2). 1 1 2 2 3 3 5 5 6 Points scored are added to the players’ totals after a grid has been completed. The game is won by the player who is first to lead by at least 10 points, having scored 100 points or more. Xavier and Yvette are playing a game of Trypples. Appropriately, Xavier is player X and Yvette is player Y. This was the grid at the end of the first round. 5 5 1 2 3 4 4 3 6 (a) (i) How many points did Xavier score in the first round? [1] (ii) How many points did Yvette score in the first round? [1]

7 © UCLES 2014 9694/32/M/J/14 In the second round, Yvette scored the maximum possible score of 38 points, whilst Xavier scored zero. (b) Draw a completed grid that would produce this result. [2] (c) Explain why it is not possible for both players to score zero in a round of Trypples. [2] In the third round Xavier placed the first tile, which was a 3, in the centre square. The players eventually drew this round 10–10, with only the middle row and the middle column scoring points. (d) Which three tiles were left in the bag at the end of the third round? Explain how you can be sure it was these three tiles. [3] This was the grid during the fourth round, after both players had placed two tiles: 1 4 6 4 Xavier won the fourth round 21–7. (e) Draw the completed fourth round grid. [3] After the sixth round, Yvette was leading with 83 points to Xavier’s 79. Currently, in the seventh round, the grid is as follows: 3 6 1 1 3 Yvette has just taken the second 6 from the bag, and has realised that she will win the game at the end of this round if she places the 6 in the top right square. (f) Explain fully how Yvette can be sure that she will win with this move. [3]

8 Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. © UCLES 2014 9694/32/M/J/14 BLANK PAGE

CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Level MARK SCHEME for the May/June 2014 series 9694 THINKING SKILLS 9694/32 Paper 3 (Problem Analysis and Solution), maximum raw mark 50 This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes should be read in conjunction with the question paper and the Principal Examiner Report for Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the May/June 2014 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components.

Page 2 Mark Scheme Syllabus Paper GCE A LEVEL – May/June 2014 9694 32 © Cambridge International Examinations 2014 1 (a) When creating a new maze, with no bungs currently in position, in how many different places can the first bung be put without blocking off any section of the maze space?[2] 7+ 7 + 9 = 23 places in total. 7 of these would block off sections of the maze (B2, B4, B5, D2, D4, F2 and F4). So answer = 16. 1 mark for identifying 6 of the places that would block off sections of the maze (implied by 17) OR for the answer 7 (the number of places which would block off a section of the maze). SC1: 11 places (correct answer for the maze shown in the QP, with the bungs in place). (b) List all the pairs of places where just two bungs would block off a section of the maze, even though neither bung on its own would. [2] B3 & C2, B3 & C4, E2 & F3, E4 & F3, C2 & C4, E2 & E4 1 mark for 4/5 correct pairs (and up to 4 incorrect) or up to 3 errors of judgment. (c) What is the greatest and least number of pockets that a maze could have, if it contains three bungs, and no areas are blocked off? [2] 1 mark for max = 10 1 mark for min = 6 If neither max nor min is correct, award 1 mark for the number of pockets calculated for any clearly-defined example. SC: 1 mark for clearly and correctly calculating max and min for two bungs: max = 11, min = 7. (d) State the starting square from which the ball cannot escape with fewer than three tilts. List three directions, in the order that they occur, that will allow the ball to escape. [2] 1 mark for B1 1 mark for any of {SE, N, SW} or {S, E, SW} or {SW, E, SW} If the correct starting point is not given, award 1 mark for a clearly-defined starting point requiring 2 tilts, with attempted directions given. (e) State where two bungs should be placed, and where the ball should start, if it is to require a route involving four tilts in order to get out. [2] 1 mark for bungs placed at D4 & D5 1 mark for E4 or F5 (accept F4) 1 mark for placing of bungs and a pathway which yields a three-tilt exit.

Page 3 Mark Scheme Syllabus Paper GCE A LEVEL – May/June 2014 9694 32 © Cambridge International Examinations 2014 2 (a) Show that Mark has left the program in state 3. Write down which state the program was in after each time Mark pressed a button. [2] 1 mark for identifying state 1 after the second press (B). 1 mark for identifying state 2 after the third press (C) and state 3 after the fourth (A). (b) What two further button presses would unlock the door? [1] C and D in the correct order. (c) Explain why the program is designed so that it will always be in state 1 after B is pressed. [1] B is not part of the code to open the door. (d) Explain why the program is designed so that pressing A while the program is in state 4 will return the program to state 3, rather than state 1. [2] State 4 means that the previous key pressed was C [1 mark]. Pressing A means that the correct first two digits have now been pressed [1 mark]. (e) Draw a table to show the next state for the program when different inputs are made while it is in states 1 to 6. [4] Input A B C D State 1 2 1 1 1 2 3 1 1 1 3 3 4 1 1 4 2 1 1 5 5 6 1 1 1 6 3 1 7 1 1 mark for the entries at A1, A2, B3, D4, A5 & C6 1 mark for entry at A4 1 mark for remainder of column A 1 mark for all the 1s

Page 4 Mark Scheme Syllabus Paper GCE A LEVEL – May/June 2014 9694 32 © Cambridge International Examinations 2014 3 (a) Suggest a simple way to choose fairly between three people, using a single roll of a die. [1] Any map that maps each person from two numbers without overlap will be acceptable. e.g. {1, 2} = A, {3, 4} = B, {5, 6} = C OR gives each person a number and ignores the rest. (b) Describe a method he could use. (Do not list all the cases.) [2] 24 ways using a 6-sided die. There are various possibilities including: The nearest edge (of 12) can be seen in two orientations (up or down). A combination of nearest edge and whether it is left or right of centre. Award 1 mark if an answer offers a way of doubling the 12 options given in the question – but is not clear how this is to be applied. (c) (i) How many different results are possible? [2] 6 6 6 6 5 4 3 5 5 4 2 3 4 5 3 2 4 2 2 3 1 1 1 1 ±2 ±1 ±3 ±4 ±1 ±2 ±1 ±2 ±4 ±3 ±1 ±2 8: –4, –3, –2, 1, 1, 2, 3, 4 Award 1 mark for a 10 (implying the inclusion of –5 & 5) or 4 (ignoring signs) or listing at least 4 correct. (ii) How many times more likely is a difference of 2 than a difference of 3? [2] Twice as likely. Award 1 mark for correctly listing the combinations which yield a difference of two (4 – 6, 3 – 1, 4 – 2 and 3 – 5) OR three (6 – 3 and 1 – 4). (d) What is the sum of opposite faces? [1] 13 (e) How can John achieve this in an unbiased way using a single roll? [3] Vertices yields multiples of five. Allocate amongst equally amongst players. Based on some unique identifier (closest point – dependent on at least 1 mark awarded). He must do something with the edges or vertices [1 mark for either] (as there are a multiple of 5 of them). There must be a way for them to be assigned uniformly to the 5 players [1 mark]. The selected edge or vertex needs to be identified (e.g. nearest at top, rightmost) [1 mark].

Page 5 Mark Scheme Syllabus Paper GCE A LEVEL – May/June 2014 9694 32 © Cambridge International Examinations 2014 (f) (i) Suggest a way for Alexander to get 60 equally likely options from one roll of a dodecahedron. [1] A combination of the face in a specific orientation (e.g. top, nearest slanting) and a specified edge or vertex (e.g. nearest vertex is to left or right). (ii) Zoe finds a way to get 120. How can she do this? [1] This could be done using the orientation (2 ways) of an edge (30 ways) combined with another feature with 2 possibilities independent of the above (e.g. the back of the top pentagon slopes to left or right). (g) Charlotte says that by rolling the standard die and the dodecahedron once each, she can fairly choose between 45 options. Suggest a way that she could achieve this. [2] 1 mark for correctly splitting 45 into 3 and 15. 1 mark for explaining how these can be achieved using two dice. 4 (a) (i) How many points did Xavier score in the first round? [1] Answer: 23 (5 + 1 + 6 and 5 + 4 + 2) (ii) How many points did Yvette score in the first round? [1] Answer: 19 (1 + 3 + 4 and 6 + 3 + 2) Allow 1 mark for Xavier = 19 AND Yvette = 23. (b) Draw a completed grid that would produce this result. [2] For example, 3 4 3 5 5 4 6 2 6 2 marks for any correct grid. A correct grid contains two 6s, 5s and 3s and one 2 AND has pairs in each row AND does not have pairs in any column. If 2 marks cannot be awarded, award 1 mark for one of the following: • a 90° rotation of a correct grid (i.e. rows scoring 38 and columns scoring 0) • a grid in which the columns score a total of 38 points • a grid in which the rows score 0

Page 6 Mark Scheme Syllabus Paper GCE A LEVEL – May/June 2014 9694 32 © Cambridge International Examinations 2014 (c) Explain why it is not possible for both players to score zero in a round of Trypples. [2] 2 marks for a clearly-expressed, correct explanation. Award 1 mark for identifying a relevant aspect, such as: • there are 6 lines (that can potentially score points) • (but) only 4 pairs of numbers can appear in the grid • A pair in each row leaves only 3 spaces to fill • x x w y y w or alternative symbolic representation z z t • if completed with pairs – the final number entered would have to be a “third digit”. (d) Which three tiles were left in the bag at the end of the third round? Explain how you can be sure it was these three tiles. [3] (The other) 3 and both 4s [1 mark] A further 2 marks, 1 for each valid element of reasoning, e.g. • the scoring lines must be 1 + 3 + 6 and 2 + 3 + 5 • 1, 2, 5 and 6 must be repeated • a valid grid: 2 1 1 2 3 5 6 6 5 (e) Draw the completed fourth round grid. [3] If 3 marks cannot be awarded, award 1 mark for completing the middle column and/or award 1 mark for any two pairs of numbers in the other two columns, provided no number appears more than twice in the grid, and/or award 1 mark for a grid in which Xavier scores 21. 2 1 5 6 4 4 6 2 5 (f) Explain fully how Yvette can be sure that she will win with this move. [3] Award 1 mark for each of the following: • Xavier could score 21 at most • Yvette will score at least 28 • Yvette will have more than 100 points with a 10-point lead