Questions - OCR D2 - A-Level Maths

Explain what a maximin route is.
Set up a dynamic programming tabulation, working backwards from stage 1, to find the two maximin routes through the network. What is the maximum load that can be transported in one journey from the warehouse to the ferry port?
A new road is opened connecting ( $2 ; 0$ ) and ( $2 ; 1$ ). There is no bridge on this road so it does not restrict the weight of the load that can be carried. Using the new road, what is the maximum load that can be transported in one journey from the warehouse to the ferry port? State the route that the delivery company should use. (Do not attempt to apply dynamic programming in this part.)

OCR D2 2006 June Q3

3 Rose and Colin repeatedly play a zero-sum game. The pay-off matrix shows the number of points won by Rose for each combination of strategies.

\multirow{6}{*}{Rose's strategy}	Colin's strategy
		$W$	$X$	$Y$	$Z$
	$A$	-1	4	-3	2
	$B$	5	-2	5	6
	C	3	-4	-1	0
	$D$	-5	6	-4	-2

What is the greatest number of points that Colin can win when Rose plays strategy $A$ and which strategy does Colin need to play to achieve this?
Show that strategy $B$ dominates strategy $C$ and also that strategy $Y$ dominates strategy $Z$. Hence reduce the game to a $3 \times 3$ pay-off matrix.
Find the play-safe strategy for each player on the reduced game. Is the game stable? Rose makes a random choice between the strategies, choosing strategy $A$ with probability $p _ { 1 }$, strategy $B$ with probability $p _ { 2 }$ and strategy $D$ with probability $p _ { 3 }$. She formulates the following LP problem to be solved using the Simplex algorithm: $$\begin{array} { l l } \text { maximise } & M = m - 5 ,
\text { subject to } & m \leqslant 4 p _ { 1 } + 10 p _ { 2 } ,
& m \leqslant 9 p _ { 1 } + 3 p _ { 2 } + 11 p _ { 3 } ,
& m \leqslant 2 p _ { 1 } + 10 p _ { 2 } + p _ { 3 } ,
& p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 ,
\text { and } & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , m \geqslant 0 . \end{array}$$ (You are not required to solve this problem.)
Explain how $9 p _ { 1 } + 3 p _ { 2 } + 11 p _ { 3 }$ was obtained. A computer gives the solution to the LP problem as $p _ { 1 } = \frac { 7 } { 48 } , p _ { 2 } = \frac { 27 } { 48 } , p _ { 3 } = \frac { 14 } { 48 }$.
Calculate the value of $M$ at this solution.

OCR D2 2006 June Q4

4 Answer this question on the insert provided. The diagram shows an activity network for a project. The table lists the durations of the activities (in hours).
\includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-05_680_1125_424_244} (ii) Key: \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e879b1f5-edc7-4819-80be-2a90dbf3d451-10_154_225_1119_1509} \captionsetup{labelformat=empty} \caption{Early event Late event time time}

\end{figure}

\includegraphics[max width=\textwidth, alt={}]{e879b1f5-edc7-4819-80be-2a90dbf3d451-10_762_1371_1409_427}

Minimum completion time = $\_\_\_\_$ hours Critical activities: $\_\_\_\_$
(iii) $\_\_\_\_$
(iv)
\includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-11_513_1189_543_520} Number of workers required = $\_\_\_\_$

(i)	$A \bullet$
$B \bullet$	$\bullet J$
$C \bullet$	$\bullet K$
$D \bullet$	$\bullet L$
$E \bullet$	$\bullet M$
$F \bullet$	$\bullet N$

(ii) $\_\_\_\_$
(iii)

	$J$	$K$	$L$	$M$	$N$	$O$
$A$	2	5	2	2	5	2
$B$	2	5	2	0	5	5
$C$	5	0	5	5	2	2
$D$
$E$
$F$

Answer part (iv) in your answer booklet.

OCR D2 2010 June Q1

1 The famous fictional detective Agatha Parrot is investigating a murder. She has identified six suspects: Mrs Lemon $( L )$, Prof Mulberry $( M )$, Mr Nutmeg $( N )$, Miss Olive $( O )$, Capt Peach $( P )$ and Rev Quince $( Q )$. The table shows the weapons that could have been used by each suspect.

Suspect
		$L$	M	$N$	$O$	$P$	$Q$
Axe handle	A		✓	✓			✓
Broomstick	$B$		✓		✓
Drainpipe	D	✓		✓
Fence post	$F$			✓	✓
Golf club	$G$	✓				✓	✓
Hammer	$H$	✓			✓	✓

Draw a bipartite graph to represent this information. Put the weapons on the left-hand side and the suspects on the right-hand side. Agatha Parrot is convinced that all six suspects were involved, and that each used a different weapon. She originally thinks that the axe handle was used by Prof Mulberry, the broomstick by Miss Olive, the drainpipe by Mrs Lemon, the fence post by Mr Nutmeg and the golf club by Capt Peach. However, this would leave the hammer for Rev Quince, which is not a possible pairing.
Draw a second bipartite graph to show this incomplete matching.
Construct the shortest possible alternating path from $H$ to $Q$ and hence find a complete matching. Write down which suspect used each weapon.
Find a different complete matching in which none of the suspects used the same weapon as in the matching from part (iii).

OCR D2 2010 June Q2

2 In an investigation into a burglary, Agatha has five suspects who were all known to have been near the scene of the crime, each at a different time of the day. She collects evidence from witnesses and draws up a table showing the number of witnesses claiming sight of each suspect near the scene of the crime at each possible time. Suspect \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Time}

		1 pm	2 pm	3 pm	4 pm	5 pm
Mrs Rowan	$R$	3	4	2	7	1
Dr Silverbirch	$S$	5	10	6	6	6
Mr Thorn	$T$	4	7	3	5	3
Ms Willow	$W$	6	8	4	8	3
Sgt Yew	$Y$	8	8	7	4	3

\end{table}

Use the Hungarian algorithm on a suitably modified table, reducing rows first, to find the matchings for which the total number of claimed sightings is maximised. Show your working clearly. Write down the resulting matchings between the suspects and the times. Further enquiries show that the burglary took place at 5 pm , and that Dr Silverbirch was not the burglar.
Who should Agatha suspect?

OCR D2 2010 June Q3

Set up a dynamic programming tabulation to find the minimum weight route from ( $0 ; 0$ ) to ( $4 ; 0$ ) on the following directed network.
\includegraphics[max width=\textwidth, alt={}, center]{406831f5-74a3-415e-8849-2c381bfe47f4-03_707_1342_1594_443} Give the route and its total weight.
Explain carefully how the route is obtained directly from the values in the table, without referring to the network.

OCR D2 2010 June Q4

4 Euan and Wai Mai play a zero-sum game. Each is trying to maximise the total number of points that they score in many repeats of the game. The table shows the number of points that Euan scores for each combination of strategies.

	Wai Mai
\cline { 2 - 5 }	$X$	$Y$	$Z$
$A$	2	- 5	3
\cline { 2 - 5 } $E u a n$	- 1	- 3	4
\cline { 1 - 5 } $C$	3	- 5	2
$D$	3	- 2	- 1

Explain what the term 'zero-sum game' means.
How many points does Wai Mai score if she chooses $X$ and Euan chooses $A$ ?
Why should Wai Mai never choose strategy $Z$ ?
Delete the column for $Z$ and find the play-safe strategy for Euan and the play-safe strategy for Wai Mai on the table that remains. Is the resulting game stable or not? State how you know. The value 3 in the cell corresponding to Euan choosing $D$ and Wai Mai choosing $X$ is changed to - 5 ; otherwise the table is unchanged. Wai Mai decides that she will choose her strategy by making a random choice between $X$ and $Y$, choosing $X$ with probability $p$ and $Y$ with probability $1 - p$.
Write down and simplify an expression for the expected score for Wai Mai when Euan chooses each of his four strategies.
Using graph paper, draw a graph showing Wai Mai's expected score against $p$ for each of Euan's four strategies and hence calculate the optimum value of $p$.

OCR D2 2010 June Q5

5 Answer this question on the insert provided. The network represents a system of irrigation channels along which water can flow. The weights on the arcs represent the maximum flow in litres per second.
\includegraphics[max width=\textwidth, alt={}, center]{406831f5-74a3-415e-8849-2c381bfe47f4-05_597_1553_479_296}

Calculate the capacity of the cut that separates $\{ S , B , C , E \}$ from $\{ A , D , F , G , H , T \}$.
Explain why neither arc $S C$ nor arc $B C$ can be full to capacity. Explain why the arcs $E F$ and $E H$ cannot both be full to capacity. Hence find the maximum flow along arc $H T$. When arc $H T$ carries its maximum flow, what is the flow along arc $H G$ ?
Show a flow of 58 litres per second on the diagram in the insert, and find a cut of capacity 58. The direction of flow in $H G$ is reversed.
Use the diagram in the insert to show the excess capacities and potential backflows for your flow from part (iii) in this case.
Without augmenting the labels from part (iv), write down flow augmenting routes to enable an additional 2 litres per second to flow from $S$ to $T$.
Show your augmented flow on the diagram in the insert. Explain how you know that this flow is maximal.

OCR D2 2010 June Q6

6 Answer parts (i), (ii) and (iii) of this question on the insert provided. The activity network for a project is shown below. The durations are in minutes. The events are numbered 1, 2, 3, etc. for reference.
\includegraphics[max width=\textwidth, alt={}, center]{406831f5-74a3-415e-8849-2c381bfe47f4-06_747_1249_482_447}

Complete the table in the insert to show the immediate predecessors for each activity.
Explain why the dummy activity is needed between event 2 and event 3, and why the dummy activity is needed between event 4 and event 5 .
Carry out a forward pass to find the early event times and a backward pass to find the late event times. Record your early event times and late event times in the table in the insert. Write down the minimum project completion time and the critical activities. Suppose that the duration of activity $K$ changes to $x$ minutes.
Find, in terms of $x$, expressions for the early event time and the late event time for event 9 .
Find the maximum duration of activity $K$ that will not affect the minimum project completion time found in part (iii). \section*{ADVANCED GCE
MATHEMATICS} Decision Mathematics 2
INSERT for Questions 5 and 6
Dummy activity is needed between event 2 and event 3 because $\_\_\_\_$
Dummy activity is needed between event 4 and event 5 because $\_\_\_\_$
Event 1 2 3 4 5 6 7 8 9 10
Early event time
Late event time
Minimum project completion time = $\_\_\_\_$ minutes Critical activities: $\_\_\_\_$ \section*{Answer part (iv) and part (v) in your answer booklet.} OCR
RECOGNISING ACHIEVEMENT

OCR D2 Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{89e3545e-fa4b-47dd-8651-7c8f998df9e7-1_762_1475_205_239} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A salesman is planning a four-day trip beginning at his home and ending at town $I$. He will spend the first night in town $A , B$ or $C$, the second night in town $D , E$ or $F$ and the third night in town $G$ or $H$. The network in Figure 1 shows the distances, in tens of miles, that he will drive each day according to the route he chooses. Use dynamic programming to find the shortest route the salesman can take and state the distance he will drive in total using this route.

OCR D2 Q2

2. Four athletes are put forward for selection for a mixed stage relay race at a local competition. They may each be selected for a maximum of one stage and only one athlete can be entered for each stage. The average time, in seconds, for each athlete to complete each stage is given below, based on past performances.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Stage
\cline { 2 - 4 } \multicolumn{1}{c\|}{}	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$
Alex	19	69	168
Darren	22	64	157
Leroy	20	72	166
Suraj	23	66	171

Use the Hungarian algorithm to find an optimal allocation which will minimise the team's total time. Your answer should show clearly how you have applied the algorithm.

OCR D2 Q3

3. A project consists of 11 activities, some of which are dependent on others having been completed. The following precedence table summarises the relevant information.

Activity	Depends on	Duration (hours)
A	-	5
B	A	4
C	A	2
D	B, C	11
E	C	4
$F$	D	3
G	D	8
$H$	D, E	2
I	$F$	1
J	$F , G , H$	7
$K$	$I , J$	2

Draw an activity network for the project.
Find the critical path and the minimum time in which the project can be completed. Activity $F$ can be carried out more cheaply if it is allocated more time.
Find the maximum time that can be allocated to activity $F$ without increasing the minimum time in which the project can be completed.

OCR D2 Q4

A sheet is provided for use in answering this question.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{89e3545e-fa4b-47dd-8651-7c8f998df9e7-3_725_1303_274_340} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 above shows a capacitated, directed network. The number on each arc indicates the capacity of that arc.

Calculate the values of cuts $C _ { 1 }$ and $C _ { 2 }$.
Find the minimum cut and state its value. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{89e3545e-fa4b-47dd-8651-7c8f998df9e7-3_645_1316_1430_338} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows a feasible flow through the same network.
State the values of $x , y$ and $z$.
Using this as your initial flow pattern, use the labelling procedure to find a maximal flow. You should list each flow-augmenting route you use together with its flow. State how you know that you have found a maximal flow.

OCR D2 Q5

5. The payoff matrix for player $X$ in a two-person zero-sum game is shown below.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	$Y$
\cline { 2 - 5 } \multicolumn{2}{c\|}{}	$Y _ { 1 }$	$Y _ { 2 }$	$Y _ { 3 }$
\multirow{2}{*}{$X$}	$X _ { 1 }$	10	4	3
\cline { 2 - 5 }	$X _ { 2 }$	${ } ^ { - } 4$	${ } ^ { - } 1$	9

Using a graphical method, find the optimal strategy for player $X$.
Find the optimal strategy for player $Y$.
Find the value of the game.

OCR D2 Q1

The payoff matrix for player $A$ in a two-person zero-sum game with value $V$ is shown below.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	$B$
\cline { 2 - 5 } \multicolumn{2}{c\|}{}	I	II	III
\multirow{3}{*}{$A$}	I	6	- 4	- 1
\cline { 2 - 5 }	II	- 2	5	3
\cline { 2 - 5 }	III	5	1	- 3

Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player $B$.

Rewrite the matrix as necessary and state the new value of the game, $v$, in terms of $V$.
Define your decision variables.
Write down the objective function in terms of your decision variables.
Write down the constraints.

OCR D2 Q2

2. The owner of a small plane is planning a journey from her local airport, $A$ to the airport nearest her parents, $K$. On the journey she will make three refuelling stops, the first at $B , C$ or $D$, the second at $E , F$ or $G$ and the third at $H , I$ or $J$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{34728928-2a21-463d-982e-c46ab2dc05c8-2_723_1303_427_356} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Figure 1 shows all the possible flights that can be made on the journey with the number by each arc indicating the distance of each flight in hundreds of miles. The owner of the plane wishes to choose a route that minimises the total distance that she flies. Use dynamic programming to find the route that she should use and state the total length of this route.

OCR D2 Q3

Four sales representatives ( $R _ { 1 } , R _ { 2 } , R _ { 3 }$ and $R _ { 4 }$ ) are to be sent to four areas ( $A _ { 1 } , A _ { 2 } , A _ { 3 }$ and $A _ { 4 }$ ) such that each representative visits one area. The estimated profit, in tens of pounds, that each representative will make in each area is shown in the table below.

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	$A _ { 1 }$	$A _ { 2 }$	$A _ { 3 }$	$A _ { 4 }$
$R _ { 1 }$	37	29	44	51
$R _ { 2 }$	45	30	43	41
$R _ { 3 }$	32	27	39	50
$R _ { 4 }$	43	25	51	55

Use the Hungarian method to obtain an allocation which will maximise the total profit made from the visits. Show the state of the table after each stage in the algorithm.

OCR D2 Q4

Execute a forward scan to find the minimum time in which the project can be completed.
Execute a backward scan to determine which activities lie on the critical path. The contractor is committed to completing the project in this minimum time and faces a penalty of $\pounds 50000$ for each day that the project is late. Unfortunately, before any work has begun, flooding means that activity $E$ will take 3 days longer than the 7 days allocated.
Activity $K$ could be completed in 1 day at an extra cost of $\pounds 90000$. Explain why doing this is not economical.
(2 marks)
If the time taken to complete any one activity, other than $E$, could be reduced by 2 days at an extra cost of $\pounds 80000$, for which activities on their own would this be profitable. Explain your reasoning.
(3 marks)
11 marks

OCR D2 Q5

A sheet is provided for use in answering this question.

A town has adopted a one-way system to cope with recent problems associated with congestion in one area. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{34728928-2a21-463d-982e-c46ab2dc05c8-5_684_1320_454_316} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} Figure 3 models the one-way system as a capacitated directed network. The numbers on the arcs are proportional to the number of vehicles that can pass along each road in a given period of time.

Find the capacity of the cut which passes through the $\operatorname { arcs } A E , B F , B G$ and $C D$.
(1 mark) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{34728928-2a21-463d-982e-c46ab2dc05c8-6_714_1280_171_333} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Figure 4 shows a feasible flow of 17 through the same network. For convenience, a supersource, $S$, and a supersink, $T$, have been used.
1. Use the labelling procedure to find the maximum flow through this network listing each flow-augmenting route you use together with its flow.
2. Show your maximum flow pattern and state its value.
Prove that your flow is the maximum possible through the network.
It is suggested that the maximum flow through the network could be increased by making road $E F$ undirected, so that it has a capacity of 8 in either direction. Using the maximum flow-minimum cut theorem, find the increase in maximum flow this change would allow.

OCR D2 Q1

Whilst Clive is in hospital, four of his friends decide to redecorate his lounge as a welcomehome surprise. They divide the work to be done into four jobs which must be completed in the following order:

strip the wallpaper,
paint the woodwork and ceiling,
hang the new wallpaper,
replace the fittings and tidy up.

The table below shows the time, in hours, that each of the friends is likely to take to complete each job.

	Alice	Bhavin	Carl	Dieter
Strip wallpaper	5	3	5	4
Paint	7	5	6	4
Hang wallpaper	8	4	7	6
Replace fittings	5	3	2	3

As they do not know how long Clive will be in hospital his friends wish to complete the redecoration in the shortest possible total time.

Use the Hungarian method to obtain the optimal allocation of the jobs, showing the state of the table after each stage in the algorithm.
Hence find the minimum time in which the friends can redecorate the lounge.

OCR D2 Q2

2. The payoff matrix for player $A$ in a two-person zero-sum game is shown below.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	$B$
\cline { 2 - 5 } \multicolumn{2}{c\|}{}	I	II	III
\multirow{3}{*}{$A$}	I	3	5	- 2
\cline { 2 - 5 }	II	7	${ } ^ { - } 4$	- 1
\cline { 2 - 5 }	III	9	${ } ^ { - } 4$	8

Applying the dominance rule, explain, with justification, which strategy can be ignored by
1. player $A$,
2. player $B$.
For the reduced table, find the optimal strategy for
1. player $A$,
2. player $B$.

OCR D2 Q3

A sheet is provided for use in answering this question.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{df7b056f-1446-43f1-a2fd-c0d56533550e-3_588_1285_287_296} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Figure 1 shows a capacitated, directed network. The numbers on each arc indicate the minimum and maximum capacity of that arc. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{df7b056f-1446-43f1-a2fd-c0d56533550e-3_648_1288_1155_296} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows a feasible flow through the same network.

Using this as your initial flow pattern, use the labelling procedure to find a maximal flow. You should list each flow-augmenting route you use together with its flow and draw the maximal flow pattern.
(6 marks)
Find a cut of the same value as your maximum flow and explain why this proves it gives the maximim possible flow.

OCR D2 Q4

Activity	Time	Precedence
A	12
B	5
C	10
D	8	A
E	5	A, B , C
F	9	C
G	11	D, E
H	6	G, F
I	6	H
J	2	H
K	3	I

Construct an activity network to show the tasks involved in widening a bridge over the B451.

Find those tasks which lie on the critical path and list them in order.
State the minimum length of time needed to widen the bridge.
Represent the tasks on a Gantt diagram. Tasks $F$ and $J$ each require 3 workers, tasks $B$, $D$ and $I$ each require 2 workers and the remaining tasks each require one worker.
Draw a resource histogram showing how it is possible for a team of 4 workers to complete the project in the minimum possible time.

OCR D2 Q5

A company wishes to plan its production of a particular item over the coming four months based on its current orders. In each month the company can manufacture up to three of the item with the costs according to how many it makes being as follows:

No. of Items Produced	0	1	2	3
Cost in Pounds	0	5500	9700	13100

There are no items in stock at the start of the period and the company wishes to meet all its orders on time and have no stock left at the end of the 4-month period. If any items are not to be supplied in the month they are made there is also a storage cost incurred of $\pounds 400$ per item per month. The orders for each of the four months being considered are as follows:

Month	March	April	May	June
No. of Orders	1	2	4	1

Use dynamic programming to find how many of the item the company should make in each of these four months in order to minimise the total cost for this period. \section*{Please hand this sheet in for marking} \includegraphics[max width=\textwidth, alt={}, center]{df7b056f-1446-43f1-a2fd-c0d56533550e-6_588_1285_504_276}
\includegraphics[max width=\textwidth, alt={}, center]{df7b056f-1446-43f1-a2fd-c0d56533550e-6_588_1280_1361_276}

OCR D2 Q1

The payoff matrix for player $A$ in a two-person zero-sum game is shown below.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	$B$
\cline { 2 - 5 } \multicolumn{2}{c\|}{}	I	II	III
\multirow{3}{*}{$A$}	I	- 3	4	0
\cline { 2 - 5 }	II	2	2	1
\cline { 2 - 5 }	III	3	- 2	- 1

the minimum number of days needed to complete the entire project,
the activities which lie on the critical path.

(i)	\(A \bullet\)
\(B \bullet\)	\(\bullet J\)
\(C \bullet\)	\(\bullet K\)
\(D \bullet\)	\(\bullet L\)
\(E \bullet\)	\(\bullet M\)
\(F \bullet\)	\(\bullet N\)

	\(J\)	\(K\)	\(L\)	\(M\)	\(N\)	\(O\)
\(A\)	2	5	2	2	5	2
\(B\)	2	5	2	0	5	5
\(C\)	5	0	5	5	2	2
\(D\)
\(E\)
\(F\)

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	\(Y\)
\cline { 2 - 5 } \multicolumn{2}{c\|}{}	\(Y _ { 1 }\)	\(Y _ { 2 }\)	\(Y _ { 3 }\)
\multirow{2}{*}{\(X\)}	\(X _ { 1 }\)	10	4	3
\cline { 2 - 5 }	\(X _ { 2 }\)	\({ } ^ { - } 4\)	\({ } ^ { - } 1\)	9

\multirow{6}{*}{Rose's strategy}	Colin's strategy
		\(W\)	\(X\)	\(Y\)	\(Z\)
	\(A\)	-1	4	-3	2
	\(B\)	5	-2	5	6
	C	3	-4	-1	0
	\(D\)	-5	6	-4	-2

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	\(A _ { 1 }\)	\(A _ { 2 }\)	\(A _ { 3 }\)	\(A _ { 4 }\)
\(R _ { 1 }\)	37	29	44	51
\(R _ { 2 }\)	45	30	43	41
\(R _ { 3 }\)	32	27	39	50
\(R _ { 4 }\)	43	25	51	55

Questions — OCR D2 (141 questions)

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