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AQA D1 2009 June Q4

13 marks Moderate -0.5

4 The diagram opposite shows a network of roads on a housing estate. The number on each edge is the length, in metres, of the road. Joe is starting a kitchen-fitting business.

Joe delivers leaflets advertising his business. He walks along all of the roads at least once, starting and finishing at $C$. Find the length of an optimal Chinese postman route for Joe.
Joe gets a job fitting a kitchen in a house at $T$. Joe starts from $C$ and wishes to drive to $T$. Use Dijkstra's algorithm on the diagram opposite to find the minimum distance to drive from $C$ to $T$. State the corresponding route. \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-08_1791_1705_916_155}
(b) \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-09_2395_1602_312_260}

AQA D1 2009 June Q5

10 marks Moderate -0.3

5 Angelo is visiting six famous places in Palermo: $A , B , C , D , E$ and $F$. He intends to travel from one place to the next until he has visited all of the places before returning to his starting place. Due to the traffic system, the time taken to travel between two places may be different dependent on the direction travelled. The table shows the times, in minutes, taken to travel between the six places.

\backslashbox{From}{To}	A	$\boldsymbol { B }$	$\boldsymbol { C }$	$\boldsymbol { D }$	E	$F$
A	-	25	20	20	27	25
$\boldsymbol { B }$	15	-	10	11	15	30
$\boldsymbol { C }$	5	30	-	15	20	19
$\boldsymbol { D }$	20	25	15	-	25	10
$\boldsymbol { E }$	10	20	7	15	-	15
F	25	35	29	20	30	-

Give an example of a Hamiltonian cycle in this context.
1. Show that, if the nearest neighbour algorithm starting from $F$ is used, the total travelling time for Angelo would be 95 minutes.
2. Explain why your answer to part (b)(i) is an upper bound for the minimum travelling time for Angelo.
Angelo starts from $F$ and visits $E$ next. He also visits $B$ before he visits $D$. Find an improved upper bound for Angelo's total travelling time.

\includegraphics[max width=\textwidth, alt={}]{44bbec2c-32f4-4d28-9dd6-d89387228454-11_2484_1709_223_153}

AQA D1 2009 June Q6

21 marks Moderate -0.3

6 Each day, a factory makes three types of widget: basic, standard and luxury. The widgets produced need three different components: type $A$, type $B$ and type $C$. Basic widgets need 6 components of type $A , 6$ components of type $B$ and 12 components of type $C$.
Standard widgets need 4 components of type $A , 3$ components of type $B$ and 18 components of type $C$.
Luxury widgets need 2 components of type $A , 9$ components of type $B$ and 6 components of type $C$.
Each day, there are 240 components of type $A$ available, 300 of type $B$ and 900 of type $C$.
Each day, the factory must use at least twice as many components of type $C$ as type $B$.
Each day, the factory makes $x$ basic widgets, $y$ standard widgets and $z$ luxury widgets.

In addition to $x \geqslant 0 , y \geqslant 0$ and $z \geqslant 0$, find four inequalities in $x , y$ and $z$ that model the above constraints, simplifying each inequality.
Each day, the factory makes the maximum possible number of widgets. On a particular day, the factory must make the same number of luxury widgets as basic widgets.
1. Show that your answers in part (a) become $$2 x + y \leqslant 60 , \quad 5 x + y \leqslant 100 , \quad x + y \leqslant 50 , \quad y \geqslant x$$
2. Using the axes opposite, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region.
3. Find the total number of widgets made on that day.
4. Find all possible combinations of the number of each type of widget made that correspond to this maximum number.

AQA D1 2009 June Q7

8 marks Moderate -0.3

The diagram shows a graph with 16 vertices and 16 edges. \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_200_204_365_758} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_200_202_365_1080} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_200_204_689_758} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_200_200_689_1082}
1. On Figure 1 below, add the minimum number of edges to make a connected graph.
2. On Figure 2 opposite, add the minimum number of edges to make the graph Hamiltonian.
3. On Figure 3 opposite, add the minimum number of edges to make the graph Eulerian.
A complete graph has $n$ vertices and is Eulerian.
1. State the condition that $n$ must satisfy.
2. The number of edges in a Hamiltonian cycle for the graph is the same as the number of edges in an Eulerian trail. State the value of $n$. \section*{Figure 1} \section*{Connected Graph} \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{(a)(i)} \includegraphics[alt={},max width=\textwidth]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_195_200_2014_758}
  \end{figure} \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{(a)(i)} \includegraphics[alt={},max width=\textwidth]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_197_200_2012_1082}
  \end{figure}
  \includegraphics[max width=\textwidth, alt={}]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_200_202_2334_758}
  \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{(a)(i)} \includegraphics[alt={},max width=\textwidth]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_200_202_2334_1080}
  \end{figure} \section*{Figure 2} \section*{Hamiltonian Graph} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-15_218_536_511_753} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-15_213_212_836_753} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-15_215_214_836_1073}
  1. (iii) \section*{Eulerian Graph} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{44bbec2c-32f4-4d28-9dd6-d89387228454-15_207_204_1356_758}
    \end{figure} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{44bbec2c-32f4-4d28-9dd6-d89387228454-15_211_206_1354_1078}
    \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-15_206_209_1681_753} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-15_200_202_1681_1080}

AQA D1 2011 June Q1

7 marks Moderate -0.5

1 Six people, $A , B , C , D , E$ and $F$, are to be allocated to six tasks, 1, 2, 3, 4, 5 and 6. The following adjacency matrix shows the tasks that each of the people is able to undertake.

	1	2	3	4	5	6
$\boldsymbol { A }$	1	0	1	0	0	0
$\boldsymbol { B }$	1	1	0	0	0	1
C	0	1	0	1	0	0
$\boldsymbol { D }$	0	1	0	0	1	0
E	0	0	1	0	1	0
F	0	0	0	0	1	0

Represent this information in a bipartite graph.
Initially, $A$ is assigned to task 3, $B$ to task 2, $C$ to task 4 and $D$ to task 5 . Use an algorithm from this initial matching to find a maximum matching, listing your alternating paths.

AQA D1 2011 June Q2

6 marks Easy -1.2

2 Five different integers are to be sorted into ascending order.

A bubble sort is to be used on the list of numbers $\quad 6 \quad 4 \quad 5 \quad x \quad 2 \quad 11$.
1. After the first pass, the list of numbers becomes $$\begin{array} { l l l l l } 4 & x & 2 & 6 & 11 \end{array}$$ Write down an inequality that $x$ must satisfy.
2. After the second pass, the list becomes $$\begin{array} { l l l l l } x & 2 & 4 & 6 & 11 \end{array}$$ Write down a new inequality that $x$ must satisfy.
The five integers are now written in a different order. A shuttle sort is to be used on the list of numbers $\quad \begin{array} { l l l l l } 11 & x & 2 & 4 & 6 . \end{array}$
1. After the first pass, the list of numbers becomes $$\begin{array} { l l l l l } x & 11 & 2 & 4 & 6 \end{array}$$ Write down an inequality that $x$ must satisfy.
2. After the second pass, the list becomes $$\begin{array} { l l l l l } 2 & x & 11 & 4 & 6 \end{array}$$ Write down a further inequality that $x$ must satisfy.
Use your answers from parts (a) and (b) to write down the value of $x$.

AQA D1 2011 June Q3

9 marks Moderate -0.8

3 A group of eight friends, $A , B , C , D , E , F , G$ and $H$, keep in touch by sending text messages. The cost, in pence, of sending a message between each pair of friends is shown in the following table.

	$\boldsymbol { A }$	$\boldsymbol { B }$	$\boldsymbol { C }$	$\boldsymbol { D }$	$\boldsymbol { E }$	$\boldsymbol { F }$	$\boldsymbol { G }$	$\boldsymbol { H }$
$\boldsymbol { A }$	-	15	10	12	16	11	14	17
$\boldsymbol { B }$	15	-	15	14	15	16	16	15
$\boldsymbol { C }$	10	15	-	11	10	12	14	9
$\boldsymbol { D }$	12	14	11	-	11	12	14	12
$\boldsymbol { E }$	16	15	10	11	-	13	15	14
$\boldsymbol { F }$	11	16	12	12	13	-	14	8
G	14	16	14	14	15	14	-	13
$\boldsymbol { H }$	17	15	9	12	14	8	13	-

One of the group wishes to pass on a piece of news to all the other friends, either by a direct text or by the message being passed on from friend to friend, at the minimum total cost.

1. Use Prim's algorithm starting from $A$, showing the order in which you select the edges, to find a minimum spanning tree for the table.
2. Draw your minimum spanning tree.
3. Find the minimum total cost.
Person $H$ leaves the group. Find the new minimum total cost.

AQA D1 2011 June Q4

9 marks Easy -1.2

4 The network below shows some pathways at a school connecting different departments. The number on each edge represents the time taken, in minutes, to walk along that pathway. Carol, the headteacher, wishes to walk from her office ( $O$ ) to the Drama department (D) .

1. Use Dijkstra's algorithm on the network to find the minimum walking time from $O$ to $D$.
2. Write down the corresponding route.
On another occasion, Carol needs to go from her office to the Business Studies department $( B )$.
1. Write down her minimum walking time.
2. Write down the route corresponding to this minimum time. \includegraphics[max width=\textwidth, alt={}, center]{3b7f04ff-e340-41ad-b50e-a02f94f02e8b-08_1499_1714_1208_153}

AQA D1 2011 June Q5

10 marks Moderate -0.5

5 A council is responsible for gritting main roads in a district. The network shows the main roads in the district. The number on each edge shows the length of the road, in kilometres. The gritter starts from the depot located at point $A$, and must drive along all the roads at least once before returning to the depot. \includegraphics[max width=\textwidth, alt={}, center]{3b7f04ff-e340-41ad-b50e-a02f94f02e8b-10_1294_923_525_555}

Find the length of an optimal Chinese postman route around the main roads in the district, starting and finishing at $A$.
Zac, a supervisor, wishes to inspect all the roads. He leaves the depot, located at point $A$, and drives along all the roads at least once before finishing at his home, located at point $C$. Find the length of an optimal route for Zac.
Liz, a reporter, intends to drive along all the roads at least once in order to report on driving conditions. She can start her journey at any point and can finish her journey at any point.
1. Find the length of an optimal route for Liz.
2. State the points from which Liz could start in order to achieve this optimal route.
  \includegraphics[max width=\textwidth, alt={}]{3b7f04ff-e340-41ad-b50e-a02f94f02e8b-11_2486_1714_221_153}

AQA D1 2011 June Q6

7 marks Easy -1.2

6 A student is tracing the following algorithm.
Line 10 Let $A = 6$ Line $20 \quad$ Let $B = 7$ Line 30 Input $C$ Line 40 Let $D = ( A + B ) / 2$ Line $50 \quad$ Let $E = C - D ^ { 3 }$ Line 60 If $E ^ { 2 } < 1$ then go to Line 120
Line 70 If $E > 0$ then go to Line 100
Line 80 Let $B = D$ Line 90 Go to Line 40
Line $100 \quad$ Let $A = D$ Line 110 Go to Line 40
Line 120 Stop

Trace the algorithm in the case where the input value is $C = 300$.
The algorithm is intended to find the approximate cube root of any input number. State two reasons why the algorithm is unsatisfactory in its present form.
(3 marks)

AQA D1 2011 June Q7

12 marks Moderate -0.8

7 A builder needs some screws, nails and plugs. At the local store, there are packs containing a mixture of the three items. A DIY pack contains 200 nails, 200 screws and 100 plugs.
A trade pack contains 1000 nails, 1500 screws and 2500 plugs.
A DIY pack costs $\pounds 2.50$ and a trade pack costs $\pounds 15$.
The builder needs at least 5000 nails, 6000 screws and 4000 plugs.
The builder buys $x$ DIY packs and $y$ trade packs and wishes to keep his total cost to a minimum.

Formulate the builder's situation as a linear programming problem.
1. On the grid opposite, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and the direction of an objective line.
2. Use your diagram to find the number of each type of pack that the builder should buy in order to minimise his cost.
3. Find the builder's minimum cost.

AQA D1 2011 June Q8

15 marks Moderate -0.8

8 Mrs Jones is a spy who has to visit six locations, $P , Q , R , S , T$ and $U$, to collect information. She starts at location $Q$, and travels to each of the other locations, before returning to $Q$. She wishes to keep her travelling time to a minimum. The diagram represents roads connecting different locations. The number on each edge represents the travelling time, in minutes, along that road. \includegraphics[max width=\textwidth, alt={}, center]{3b7f04ff-e340-41ad-b50e-a02f94f02e8b-16_524_866_612_587}

1. Explain why the shortest time to travel from $P$ to $R$ is 40 minutes.
2. Complete Table 1, on the opposite page, in which the entries are the shortest travelling times, in minutes, between pairs of locations.
1. Use the nearest neighbour algorithm on Table 1, starting at $Q$, to find an upper bound for the minimum travelling time for Mrs Jones's tour.
2. Mrs Jones decides to follow the route given by the nearest neighbour algorithm. Write down her route, showing all the locations through which she passes.

By deleting $Q$ from Table 1, find a lower bound for the travelling time for Mrs Jones's tour. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 1}

	$\boldsymbol { P }$	$Q$	$\boldsymbol { R }$	$\boldsymbol { S }$	$T$	$\boldsymbol { U }$
$P$	-	25
$Q$	25	-	20	21	23	11
$\boldsymbol { R }$		20	-
$\boldsymbol { S }$		21		-
$T$		23			-	12
$\boldsymbol { U }$		11			12	-

\end{table}

\includegraphics[max width=\textwidth, alt={}]{3b7f04ff-e340-41ad-b50e-a02f94f02e8b-18_2486_1714_221_153}

AQA D1 2012 June Q1

5 marks Easy -1.2

1 Six people, $A , B , C , D , E$ and $F$, are to be allocated to six tasks, 1, 2, 3, 4, 5 and 6. The following bipartite graph shows the tasks that each of the people is able to undertake. \includegraphics[max width=\textwidth, alt={}, center]{1258a6d3-558a-46dc-a916-d71f71b175ff-02_1003_547_740_737}

Represent this information in an adjacency matrix.
Initially, $B$ is assigned to task 4, $C$ to task 3, $D$ to task 1, $E$ to task 5 and $F$ to task 6. By using an algorithm from this initial matching, find a complete matching.
(3 marks)

AQA D1 2012 June Q2

5 marks Easy -1.8

2 A student is using a shuttle sort algorithm to rearrange a set of numbers into ascending order. Her correct solution for the first three passes is as follows.

AQA D1 2012 June Q3

9 marks Easy -1.3

3 The following network shows the lengths, in miles, of roads connecting nine villages, $A , B , \ldots , I$. \includegraphics[max width=\textwidth, alt={}, center]{1258a6d3-558a-46dc-a916-d71f71b175ff-06_810_501_445_388} \includegraphics[max width=\textwidth, alt={}, center]{1258a6d3-558a-46dc-a916-d71f71b175ff-06_812_499_443_1135}

1. Use Prim's algorithm starting from $A$, showing the order in which you select the edges, to find a minimum spanning tree for the network.
2. State the length of your minimum spanning tree.
3. Draw your minimum spanning tree.
Prim's algorithm from different starting points produces the same minimum spanning tree for this network. State the final edge that would complete the minimum spanning tree using Prim's algorithm:
1. starting from $D$;
2. starting from $H$.

AQA D1 2012 June Q4

8 marks Easy -1.2

4 The edges on the network below represent some major roads in a city. The number on each edge is the minimum time taken, in minutes, to drive along that road.

1. Use Dijkstra's algorithm on the network to find the shortest possible driving time from $A$ to $J$.
2. Write down the corresponding route.
A new ring road is to be constructed connecting $A$ to $J$ directly. Find the maximum length of this new road from $A$ to $J$ if the time taken to drive along it, travelling at an average speed of $90 \mathrm {~km} / \mathrm { h }$, is to be no more than the time found in part (a)(i). \section*{(a)(i)} \includegraphics[max width=\textwidth, alt={}, center]{1258a6d3-558a-46dc-a916-d71f71b175ff-08_912_1276_1053_429}

AQA D1 2012 June Q5

8 marks Standard +0.3

5 The network below shows some streets in a town. The number on each edge shows the length of that street, in metres. Leaflets are to be distributed by a restaurant owner, Tony, from his restaurant located at vertex $B$. Tony must start from his restaurant, walk along all the streets at least once, before returning to his restaurant. \includegraphics[max width=\textwidth, alt={}, center]{1258a6d3-558a-46dc-a916-d71f71b175ff-10_643_1353_625_340} The total length of the streets is 2430 metres.

Find the length of an optimal Chinese postman route for Tony.
Colin also wishes to distribute some leaflets. He starts from his house at $H$, walks along all the streets at least once, before finishing at the restaurant at $B$. Colin wishes to walk the minimum distance. Find the length of an optimal route for Colin.
David also walks along all the streets at least once. He can start at any vertex and finish at any vertex. David also wishes to walk the minimum distance.
1. Find the length of an optimal route for David.
2. State the vertices from which David could start in order to achieve this optimal route.

AQA D1 2012 June Q6

7 marks Moderate -0.8

6 The complete graph $K _ { n } ( n > 1 )$ has every one of its $n$ vertices connected to each of the other vertices by a single edge.

Draw the complete graph $K _ { 4 }$.
1. Find the total number of edges for the graph $K _ { 8 }$.
2. Give a reason why $K _ { 8 }$ is not Eulerian.
For the graph $K _ { n }$, state in terms of $n$ :
1. the total number of edges;
2. the number of edges in a minimum spanning tree;
3. the condition for $K _ { n }$ to be Eulerian;
4. the condition for the number of edges of a Hamiltonian cycle to be equal to the number of edges of an Eulerian cycle.

AQA D1 2012 June Q7

11 marks Moderate -0.8

7 Rupta, a sales representative, has to visit six shops, $A , B , C , D , E$ and $F$. Rupta starts at shop $A$ and travels to each of the other shops once, before returning to shop $A$. Rupta wishes to keep her travelling time to a minimum. The table shows the travelling times, in minutes, between the shops.

AQA D1 2012 June Q8

8 marks Easy -1.2

8 The following algorithm finds an estimate of the value of the number represented by the symbol e:

Line 10	Let $A = 1 , B = 1 , C = 1$
Line 20	Let $D = A$
Line 30	Let $C = C \times B$
Line 40	Let $D = D + ( 1 / C )$
Line 50	If $B = 4$ then go to Line 80
Line 60	Let $B = B + 1$
Line 70	Go to Line 30
Line 80	Print 'An estimate of e is', $D$
Line 90	End

Trace the algorithm.
A student miscopied Line 70 . His line was
Line 70 Go to Line 10
Explain what would happen if his algorithm were traced.

AQA D1 2012 June Q9

14 marks Moderate -0.3

9 Ollyin is buying new pillows for his hotel. He buys three types of pillow: soft, medium and firm. He must buy at least 100 soft pillows and at least 200 medium pillows.
He must buy at least 400 pillows in total.
Soft pillows cost $\pounds 4$ each. Medium pillows cost $\pounds 3$ each. Firm pillows cost $\pounds 4$ each.
He wishes to spend no more than $\pounds 1800$ on new pillows.
At least $40 \%$ of the new pillows must be medium pillows.
Ollyin buys $x$ soft pillows, $y$ medium pillows and $z$ firm pillows.

In addition to $x \geqslant 0 , y \geqslant 0$ and $z \geqslant 0$, find five inequalities in $x , y$ and $z$ that model the above constraints.
Ollyin decides to buy twice as many soft pillows as firm pillows.
1. Show that three of your answers in part (a) become $$\begin{aligned} 3 x + 2 y & \geqslant 800 \\ 2 x + y & \leqslant 600 \\ y & \geqslant x \end{aligned}$$
2. On the grid opposite, draw a suitable diagram to represent Ollyin's situation, indicating the feasible region.
3. Use your diagram to find the maximum total number of pillows that Ollyin can buy.
4. Find the number of each type of pillow that Ollyin can buy that corresponds to your answer to part (b)(iii).
  \includegraphics[max width=\textwidth, alt={}]{1258a6d3-558a-46dc-a916-d71f71b175ff-20_2256_1707_221_153}

AQA D1 2013 June Q1

7 marks Moderate -0.8

1 Six people, Andy, Bob, Colin, Dev, Eric and Faisal, are to be allocated to six tasks, $1,2,3,4,5$ and 6 . The following table shows the tasks that each person is able to undertake.

Person	Task
Andy	1,3
Bob	1,4
Colin	2,3
Dev	$4,5,6$
Eric	$2,5,6$
Faisal	1,3

Represent this information on a bipartite graph.
Initially, Bob is allocated to task 1, Colin to task 3, Dev to task 5 and Eric to task 2. Demonstrate, by using an alternating path algorithm from this initial matching, how each person can be allocated to a different task.

AQA D1 2013 June Q2

5 marks Easy -1.8

Use the quicksort algorithm to rearrange the following numbers into ascending order, showing the new arrangement after each pass. You must indicate the pivot(s) being used on each pass. $$2 , \quad 12 , \quad 17 , \quad 18 , \quad 5 , \quad 13$$
For the first pass, write down the number of comparisons.

AQA D1 2013 June Q3

9 marks Moderate -0.5

3 The following network shows the lengths, in miles, of roads connecting ten villages, $A , B , C , \ldots , J$. \includegraphics[max width=\textwidth, alt={}, center]{77c4efd4-a905-48f3-a6f7-36b0e47dbc6d-06_899_1458_397_285}

1. Use Kruskal's algorithm, showing the order in which you select the edges, to find a minimum spanning tree for the network.
2. Find the length of your minimum spanning tree.
3. Draw your minimum spanning tree.
Prim's algorithm from different starting points produces the same minimum spanning tree. State the final edge that would be added to complete the minimum spanning tree if the starting point were:
1. $A$;
2. $F$.

AQA D1 2013 June Q4

12 marks Moderate -0.8

4 Sarah is a mobile hairdresser based at $A$. Her day's appointments are at five places: $B , C , D , E$ and $F$. She can arrange the appointments in any order. She intends to travel from one place to the next until she has visited all of the places, starting and finishing at $A$. The following table shows the times, in minutes, that it takes to travel between the six places.

\cline { 2 - 7 } \multicolumn{1}{c\|}{}	$\boldsymbol { A }$	$\boldsymbol { B }$	$\boldsymbol { C }$	$\boldsymbol { D }$	$\boldsymbol { E }$	$\boldsymbol { F }$
$\boldsymbol { A }$	-	15	11	14	27	12
$\boldsymbol { B }$	15	-	13	19	24	15
$\boldsymbol { C }$	11	13	-	10	19	12
$\boldsymbol { D }$	14	19	10	-	26	15
$\boldsymbol { E }$	27	24	19	26	-	27
$\boldsymbol { F }$	12	15	12	15	27	-

Sarah decides to visit the places in the order $A B C D E F A$. Find the travelling time of this tour.
Explain why this answer can be considered as being an upper bound for the minimum travelling time of Sarah's tour.
Use the nearest neighbour algorithm, starting from $A$, to find another upper bound for the minimum travelling time of Sarah's tour.
By deleting $A$, find a lower bound for the minimum travelling time of Sarah's tour.
Sarah thinks that she can reduce her travelling time to 75 minutes. Explain why she is wrong.

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\backslashbox{From}{To}	A	\(\boldsymbol { B }\)	\(\boldsymbol { C }\)	\(\boldsymbol { D }\)	E	\(F\)
A	-	25	20	20	27	25
\(\boldsymbol { B }\)	15	-	10	11	15	30
\(\boldsymbol { C }\)	5	30	-	15	20	19
\(\boldsymbol { D }\)	20	25	15	-	25	10
\(\boldsymbol { E }\)	10	20	7	15	-	15
F	25	35	29	20	30	-

	1	2	3	4	5	6
\(\boldsymbol { A }\)	1	0	1	0	0	0
\(\boldsymbol { B }\)	1	1	0	0	0	1
C	0	1	0	1	0	0
\(\boldsymbol { D }\)	0	1	0	0	1	0
E	0	0	1	0	1	0
F	0	0	0	0	1	0

	\(\boldsymbol { P }\)	\(Q\)	\(\boldsymbol { R }\)	\(\boldsymbol { S }\)	\(T\)	\(\boldsymbol { U }\)
\(P\)	-	25
\(Q\)	25	-	20	21	23	11
\(\boldsymbol { R }\)		20	-
\(\boldsymbol { S }\)		21		-
\(T\)		23			-	12
\(\boldsymbol { U }\)		11			12	-

Line 10	Let \(A = 1 , B = 1 , C = 1\)
Line 20	Let \(D = A\)
Line 30	Let \(C = C \times B\)
Line 40	Let \(D = D + ( 1 / C )\)
Line 50	If \(B = 4\) then go to Line 80
Line 60	Let \(B = B + 1\)
Line 70	Go to Line 30
Line 80	Print 'An estimate of e is', \(D\)
Line 90	End

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