7. For APPLICANTS IN $\left\{ \begin{array} { l } \text { COMPUTER SCIENCE } \\ \text { COMPUTER SCIENCE \& PHILOSOPHY } \end{array} \right\}$ ONLY.

You are given two identical black boxes, each with an $N$-digit display and each with two buttons marked $A$ and $B$. Button $A$ resets the display to 0 , and button $B$ updates the display using a complex, unknown but fixed, function $f$, so that pressing button $A$ then repeatedly pressing button $B$ displays a fixed sequence
$$x _ { 0 } = 0 , x _ { 1 } = f \left( x _ { 0 } \right) , x _ { 2 } = f \left( x _ { 1 } \right) , \ldots ,$$
the same for both boxes. In general $x _ { i } = f ^ { i } ( 0 )$ where $f ^ { i }$ denotes applying function $f$ repeatedly $i$ times.
You have no pencil and paper, and the display has too many digits for you to remember more than a few displayed values, but you can compare the displays on the two boxes to see if they are equal, and you can count the number of times you press each button.
(i) Explain briefly why there must exist integers $i , j$ with $0 \leqslant i < j$ such that $x _ { i } = x _ { j }$.
(ii) Show that if $x _ { i } = x _ { j }$ then $x _ { i + s } = x _ { j + s }$ for any $s \geqslant 0$.
(iii) Let $m$ be the smallest number such that $x _ { m }$ appears more than once in the sequence, and let $p > 0$ be the smallest number such that $x _ { m } = x _ { m + p }$. Show that if $i \geqslant m$ and $k \geqslant 0$ then $x _ { i + k p } = x _ { i }$.
(iv) Given integers $i , j$ with $0 \leqslant i < j$, show that $x _ { i } = x _ { j }$ if and only if $i \geqslant m$ and $j - i$ is a multiple of $p$. [Hint: let $r$ be the remainder on dividing $j - i$ by $p$, and argue that $r = 0$.]
(v) You conduct an experiment where (after resetting both boxes) you repeatedly press button $B$ once on one box and button $B$ twice on the other box and compare the displays, thus determining the smallest number $u > 0$ such that $x _ { u } = x _ { 2 u }$. What relates the value of $u$ to the (unknown) values of $m$ and $p$ ?
(vi) Once $u$ is known, what experiment would you perform to determine the value of $m$ ?
(vii) Once $u$ and $m$ are known, what experiment would tell you the value of $p$ ?
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7. For APPLICANTS IN $\left\{ \begin{array} { l } \text { COMPUTER SCIENCE } \\ \text { COMPUTER SCIENCE \& PHILOSOPHY } \end{array} \right\}$ ONLY.

Quantiles is game for a single player. It is played with an inexhaustible supply of tiles, each bearing one of the symbols $\mathrm { A } , \mathrm { B }$ or C . In each move, the player lays down a row of tiles containing exactly one A and one B , but varying numbers of C 's. The rules are as follows:

• The player may play the basic rows CACBCC and CCACBCCC.
• If the player has already played rows of the form $r \mathrm {~A} s \mathrm {~B} t$ and $x \mathrm {~A} y \mathrm {~B} z$ (they may be the same row), where each of $r , s , t , x , y , z$ represents a sequence of C's, then he or she may add the row $r x \mathrm {~A} s y \mathrm {~B} t z$, in which copies of the sequences of C's from the previous rows are concatenated with an intervening A and B : this is called a join move. The original rows remain, and may be used again in subsequent join moves.
• No other rows may be played.

The player attempts to play one row after another so as to finish with a specified goal row.
(i) Give two examples of rows, other than basic rows, that may be played in the game.
(ii) Give two examples of rows, each containing exactly one A and one B , that may never be played, and explain why.
(iii) Let $\mathrm { C } ^ { n }$ denote an unbroken sequence of $n$ tiles each labelled with C . Can the goal row $\mathrm { C } ^ { 64 } \mathrm { AC } ^ { 48 } \mathrm { BC } ^ { 112 }$ be achieved? Justify your answer.
(iv) Can the goal row $\mathrm { C } ^ { 128 } \mathrm { AC } ^ { 48 } \mathrm { BC } ^ { 176 }$ be achieved? Justify your answer.
(v) The goal row $\mathrm { C } ^ { 31 } \mathrm { AC } ^ { 16 } \mathrm { BC } ^ { 47 }$ is achievable; show that it can be reached with 7 join moves.
(vi) In any game, we call a row useless if it repeats an earlier row or it is not used in a subsequent join move. What is the maximum number of join moves in a game that ends with $\mathrm { C } ^ { 31 } \mathrm { AC } ^ { 16 } \mathrm { BC } ^ { 47 }$ and contains no useless rows?
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7. For APPLICANTS IN $\left\{ \begin{array} { l } \text { COMPUTER SCIENCE } \\ \text { COMPUTER SCIENCE \& PHILOSOPHY } \end{array} \right\}$ ONLY.

Throughout this question, all functions will be Boolean functions of Boolean input variables. A Boolean variable can be either 0 or 1 . A Boolean function may have one or more Boolean input variables, and the output of a Boolean function is also either 0 or 1 . Three elementary Boolean functions are defined as follows:

• The function $\min \left( x _ { 1 } , \ldots , x _ { k } \right)$ can take any number of inputs. It outputs the value 1 exactly when each of its inputs is 1 , that is the output of the function is the minimum value among its inputs.
• The function $\max \left( x _ { 1 } , \ldots , x _ { k } \right)$ can take any number of inputs. It outputs the value 1 exactly when at least one of its inputs is 1 , that is the output of the function is the maximum value among its inputs.
• The function flip takes a single input and is defined as flip $\left( x _ { 1 } \right) = 1 - x _ { 1 }$.

First we will consider Boolean functions obtained by combining the three elementary Boolean functions. One such function is shown below:
$$f \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) = \min \left( \max \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) , \operatorname { flip } \left( \min \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) \right) \right) .$$
(i) Describe in words when the function $f$ outputs 1 and when it outputs 0 .
(ii) The function majority $\left( x _ { 1 } , \ldots , x _ { k } \right)$ takes $k$ inputs and outputs 1 exactly when strictly greater than $k / 2$ of its inputs are 1 . Explain how you could combine elementary Boolean functions to obtain the following functions:
(a) majority $\left( x _ { 1 } , x _ { 2 } \right)$
(b) majority $\left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right)$
Now we will consider Boolean functions that can be obtained by combining only majority functions.
(iii) There are exactly 16 distinct Boolean functions of two input variables. Some of these can be represented using only majority functions that take 3 inputs; the use of 0 or 1 as fixed inputs to majority is permitted. For example, majority $\left( x _ { 1 } , x _ { 2 } , 1 \right)$ represents the function $\max \left( x _ { 1 } , x _ { 2 } \right)$. Find any four other Boolean functions of two variables that can be represented by combining one or more majority functions of 3 inputs. Write your answers in terms of majority functions.
(iv) Give an example of a Boolean function $g$ of two input variables that cannot be represented by combining majority functions (of any number of inputs). You should write your answer by explicitly specifying $g ( 0,0 ) , g ( 0,1 ) , g ( 1,0 )$ and $g ( 1,1 )$. Justify your answer.
In the last part, you may express Boolean functions by combining any of the elementary Boolean functions or the majority function.
(v) Consider four input variables $x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 }$. Let $z _ { 1 } = \min \left( x _ { 1 } , x _ { 2 } \right) , z _ { 2 } = \min \left( x _ { 2 } , x _ { 3 } \right)$, $z _ { 3 } = \min \left( x _ { 3 } , x _ { 4 } \right) , z _ { 4 } = \min \left( x _ { 4 } , x _ { 1 } \right)$. It is sometimes possible to represent a function $s \left( x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } \right)$ using a function $t \left( z _ { 1 } , z _ { 2 } , z _ { 3 } , z _ { 4 } \right)$. For example, $\min \left( x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } \right) = \min \left( z _ { 1 } , z _ { 2 } , z _ { 3 } , z _ { 4 } \right)$, as both functions output 1 if and only if all four $x _ { i }$ are 1 . Can you represent the following functions of inputs $x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 }$ as some Boolean function of inputs $z _ { 1 } , z _ { 2 } , z _ { 3 } , z _ { 4 }$ ? Justify your answers.
(a) majority $\left( x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } \right)$.
(b) The function parity $\left( x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } \right)$ which outputs 1 exactly when an odd number of its inputs are 1 .
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5. For ALL APPLICANTS.

Alice is participating in a TV game show where $n$ distinct items are placed behind $n$ closed doors. The game proceeds as follows. Alice picks a door $i$ which is opened and the item behind it is revealed. Then the door is shut again and the host secretly swaps the item behind door $i$ with the item behind one of the neighbouring doors, $i - 1$ or $i + 1$. If Alice picks door 1, the host has to then swap the item with the one behind door 2; similarly, if Alice picks door $n$, the host has to swap the item with the one behind door $n - 1$. Alice then gets to pick any door again, and the process repeats for a certain fixed number of rounds. At the end of the game, Alice wins all the items that were revealed to her.
As a concrete example, suppose $n = 3$, and if the original items behind the three doors were ( $a _ { 1 } , a _ { 2 } , a _ { 3 }$ ), then if first Alice picks door 2 , the arrangement after the host has swapped items could be either $\left( a _ { 2 } , a _ { 1 } , a _ { 3 } \right)$ or $\left( a _ { 1 } , a _ { 3 } , a _ { 2 } \right)$. So if Alice was allowed to pick twice, had she chosen door 2 followed by door 1, in the former case she would only get the item $a _ { 2 }$, whereas in the latter she would get items $a _ { 2 }$ and $a _ { 1 }$. Alice's aim is to find a sequence of door choices that guarantee her winning a large number of items, no matter how the swaps were performed.
(i) For $n = 13$, give an increasing sequence of length 7 of distinct doors that Alice can pick that guarantees she wins 7 items.
(ii) For any $n$ of the form $2 k + 1$, give a strategy to pick an increasing sequence of $k + 1$ distinct doors that Alice can use to guarantee that she wins $k + 1$ items. Briefly justify your answer.
(iii) For $n = 13$, give a sequence of length 10 of doors that Alice can pick that guarantees she wins 10 items.
(iv) For any $n$ of the form $3 k + 1$, give a strategy to pick a sequence of $2 k + 2$ doors that Alice can use to guarantee that she wins $2 k + 2$ items. Briefly justify your answer.
(v) (a) For $n = 3$, give a sequence of length 3 of doors that Alice can pick that guarantees she wins all 3 items.
(b) For $n = 5$, give a sequence of length 5 of doors that Alice can pick that guarantees she wins all 5 items.
(vi) For $n = 13$, give a sequence of length 11 of doors that Alice can pick that guarantees she wins 11 items.
(vii) For any $n$ of the form $4 k + 1$, give a strategy to pick a sequence of $3 k + 2$ doors that Alice can use to guarantee that she wins $3 k + 2$ items. Briefly justify your answer.
(viii) For $n = 6$, is there a sequence of any length of doors that Alice can choose that will guarantee that she wins all 6 items? Justify your answer.
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6. For APPLICANTS IN $\left\{ \begin{array} { l } \text { COMPUTER SCIENCE } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \\ \text { COMPUTER SCIENCE \& PHILOSOPHY } \end{array} \right\}$ ONLY.

This question is about influencer networks. An influencer network consists of $n$ influencers denoted by circles, and arrows between them. Throughout this question, each influencer holds one of two opinions, represented by either a - or a □ in the circle. We say that an influencer $A$ follows influencer $B$ if there is an arrow from $B$ to $A$; this indicates that $B$ has ability to influence $A$.
[Figure]
Figure 1
[Figure]
Figure 2
The example in Figure 1 above shows a network with $A$ and $B$ following each other, $B$ and $C$ following each other, and $C$ also following $A$. In this example, initially $B$ and $C$ have opinion □ , while $A$ has opinion $\triangle$. An influencer will change their mind according to the strict majority rule, that is, they change their opinion if strictly more than half of the influencers they are following have an opinion different from theirs. Opinions in an influence network change in rounds. In each round, each influencer will look at the influencers they are following and simultaneously change their opinion at the end of the round according to the strict majority rule. In the above network, after one round, $A$ changes their opinion because the only influencer they are following, $B$, has a differing opinion, and the network becomes as shown in Figure 2 above.
An influencer network with an initial set of opinions is stable if no influencer changes their opinion, and a network (with initial opinions) is eventually stable if after a finite number of rounds it becomes stable. The network in the above example is eventually stable as it becomes stable after one round.
(i) A network of three influencers (without opinions) is shown below. Is this influencer network eventually stable regardless of the initial opinions of the influencers $A$, $B$ and $C$ ? Justify your answer. [Figure]
(ii) Another network of influencers (without opinions) is shown below. Is this influencer network eventually stable regardless of the initial opinions of the influencers? Justify your answer.
[Figure]
( $\triangle ) A$
(iii) A partial network of influencers (without opinions for $B _ { 1 } , \ldots , B _ { 8 }$ ) is shown below. You can add at most six additional influencers, assign any opinion of your choice to the new influencers, and add any arrows to the network to describe follower relationships. Design a network that is eventually stable regardless of initial opinions, and has the property that when it becomes stable $A$ has opinion □ if and only if each of $B _ { 1 } , B _ { 2 } , \ldots , B _ { 8 }$ had opinion □ at the start. Justify your answer.
( $\triangle ) A$ [Figure]
(iv) You are given two influencer networks, $N _ { 1 }$ and $N _ { 2 }$, with disjoint sets of influencers shown below. Both are eventually stable. Suppose one of the influencers from network $N _ { 2 }$ follows the influencer $X$ from the network $N _ { 1 }$. Is the resulting network guaranteed to be eventually stable? Justify your answer. [Figure] [Figure]
(v) (a) Given a network with $n$ influencers, where the arrows are fixed, but you are allowed to assign opinions ( $\triangle$ or $\square$ ) to each influencer, how many possible assignments of opinions is possible?
(b) Given an influencer network and an initial assignment of opinions, explain how you would determine whether the influencer network is eventually stable. Justify your answer.
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7. For APPLICANTS IN $\left\{ \begin{array} { l } \text { COMPUTER SCIENCE } \\ \text { COMPUTER SCIENCE \& PHILOSOPHY } \end{array} \right\}$ ONLY.

A data operator is receiving tokens one by one through an input channel. The data operator will receive a total of $n$ tokens, where $n \geqslant 6$, and these are numbered as $1,2 , \ldots , n$. The data operator is required to pass these tokens to the output channel; however they must do so using a valid sequence. A valid sequence is one where all the odd tokens appear first, followed by the even tokens, or the other way round; furthermore all the odd tokens must appear in either increasing or decreasing order, and likewise all the even tokens must appear in increasing or decreasing order. All valid sequences for $n = 6$ are listed below.

135246	135642	531246	531642
246135	246531	642135	642531

The data operator has a storage unit that can hold a sequence, and can perform the following operations as they receive the tokens one by one.

• pass: Input token goes straight to the output channel.
• pop : Instead of using a token from the input, the token from the right end of the storage unit is removed (provided one exists) and sent to the output channel.
• pushL : Input token is pushed in at the left end of the storage unit.
• pushR : Input token is pushed in at the right end of the storage unit.

As an illustrative example, when $n = 6$, the storage and output channel are shown for the following sequence of operations, which results in the valid output sequence 13 5642.

Input Token	Operation	Storage	Output
1	pass	[ ]	1
2	pushR	$[ 2 ]$	1
3	pass	$[ 2 ]$	13
4	pushR	$[ 24 ]$	13
5	pass	$[ 24 ]$	135
6	pass	$[ 24 ]$	1356
	pop	$[ 2 ]$	13564
	pop	[ ]	135642

(i) For $n = 6$, which valid sequences can the data operator achieve?
(ii) For $n \geqslant 6$ and even, how many valid sequences are there? Justify your answer.
(iii) For $n \geqslant 6$ and even, how many valid sequences can be achieved by the data operator? Briefly justify your answer.
(iv) In the remainder of the question, $n \geqslant 9$ is a multiple of 3 . A 3 -valid output sequence is one where among the tokens $1,2 , \ldots , n$, all tokens of the form $3 k$ appear together in increasing or decreasing order, likewise all tokens of the form $3 k + 1$ appear together in increasing or decreasing order, and the same is the case for all tokens of the form $3 k + 2$. As examples, the two sequences on the left below are 3 -valid, whereas the two on the right are not - the first because it mixes groups and the second because although the groups are separate, the tokens of the form $3 k + 2$ are in neither increasing nor decreasing order.

3 -valid	not 3 -valid
147963852	135642789
258147369	285963147

For $n \geqslant 9$ and multiple of 3 , how many 3 -valid sequences of length $n$ are there? Justify your answer.
(v) For $n \geqslant 9$ and multiple of 3, given the input sequence of tokens $1,2 , \ldots , n$, how many 3 -valid sequences can be achieved by the operator using a single storage unit and the operations pass, pop, pushL and pushR? Justify your answer.
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6.
For Oxford applicants in Computer Science / Mathematics \& Computer Science / Computer Science \& Philosophy ONLY.
In an octatree, all the digits 1 to 8 are arranged in a diagram like trees $T _ { 1 }$ and $T _ { 2 }$ shown below. There is a single digit at the root, drawn at the top (so the root is 3 in $T _ { 1 }$ ), and every other digit has another digit as its parent, so that by moving up the tree from parent to parent, each non-root digit has a unique path to the root. The order in which the children of any parent are drawn does not matter, so for simplicity we show them in increasing order from left to right.
A leaf is a digit that is not the parent of any other digit: in tree $T _ { 1 }$, the leaves are $2,4,6$ and 7 .
$T _ { 1 }$ [Figure]
$T _ { 2 }$ [Figure]
The code for an octatree is a sequence of seven digits obtained as follows. We use $T _ { 1 }$ as an example.

• Remove the numerically smallest leaf and write down its parent. In $T _ { 1 }$, we remove 2 and write down its parent 8 .
• In the tree that remains, remove the smallest leaf and write down its parent. In $T _ { 1 }$, after having removed 2, we remove 4 and write down its parent 5 .
• Continue in this way until only the root remains. In $T _ { 1 }$, we would have deleted the digits $2,4,5,6,7,1,8$ in that order and obtained the code 8538183.
  (i) $[ 1$ mark $]$ Find the code for the octatree $T _ { 2 }$. [0pt] (ii) [1 mark] Draw the octatree that has the code $\mathbf { 8 8 8 8 8 8 8 }$. [0pt] (iii) [2 marks] Draw the octatree that has the code $\mathbf { 3 1 6 5 4 7 2 }$. [0pt] (iv) [3 marks] What are the leaves of the octatree that has the code $\mathbf { 1 6 1 8 3 8 8 }$ ? Justify your answer. [0pt] (v) [2 marks] Find all the digits in the octatree that has the code $\mathbf { 1 6 1 8 3 8 8 }$ that have 1 as their parent. [0pt] (vi) [2 marks] Reconstruct the whole tree that has the code 1618388. [0pt] (vii) [2 marks] Briefly describe a procedure that given a sequence of seven digits from 1 to 8 constructs an octatree with that sequence as its code. [0pt] (viii) [2 marks] Is the number of distinct octatrees greater than or smaller than $2,000,000$ ? Justify your answer. (You may use the fact that $2 ^ { 10 } = 1024$.)

The faces of a cube are each painted either red or blue.
An ant is positioned on each of the eight corners of the cube, and each ant can only see the three faces that meet at that corner of the cube.
In this question, the ants will be asked about the faces that they can see, and the ants will always answer truthfully.
(i) The ants are each asked "Can you see an even number of red faces?". Each ant answers either yes or no. Explain why the number of ants that say yes is even.
(ii) Is it possible that all eight of the ants can each see exactly two red faces? Justify your answer.
(iii) The ants are each asked "Can you see at least one red face?". Explain why it is impossible for exactly five of the ants to say yes and exactly three to say no.
(iv) Suppose that the four ants on the corners of the top face of the cube can see exactly $0,1,1$, and 2 red faces each, in some order. How many blue faces might there be in total? Find all possibilities, and explain your answer.
(v) A three-dimensional shape is constructed such that each face is either a square or a hexagon, with two faces meeting at each edge and three faces meeting at each corner. Each face of the shape is painted either red or blue. Consider the edges where a red face meets a blue face. Explain why the number of such edges is even.

Prove that $$( f ( x ) \cdot g ( x ) ) \cdot h ( x ) = f ( x ) \cdot ( g ( x ) \cdot h ( x ) )$$ for all linear polynomials $f ( x )$ and $g ( x )$ and $h ( x )$.

Given that $f ( x )$ and $g ( x )$ are linear polynomials and $f ( x ) \cdot g ( x ) = 0$, describe all possibilities for the pair $f ( x )$ and $g ( x )$.

Given that $f ( x )$ and $g ( x )$ and $h ( x )$ are linear polynomials and $$f ( x ) \cdot g ( x ) \cdot h ( x ) = 0$$ prove that at least one of the following statements must be true; (I) $f ( x ) \cdot g ( x ) = 0$, (II) $g ( x ) \cdot h ( x ) = 0$, (III) $f ( x ) \cdot h ( x ) = 0$. For each of the three statements, give examples of polynomials for which that statement is true and the other two statements are false.

Is the set $\{ 1,2,4,5,6,9,10,11 \}$ nice? Justify your answer.

Prove that for any nice set of six numbers, the total of those six numbers must be a multiple of 3.

Now we consider infinite sets of whole numbers (for example, the set of all the positive whole numbers). Give examples to demonstrate that an infinitely large set of whole numbers might have zero, exactly one, or more than one target(s). Justify your answers, making it clear which is which.

Suppose that we want to find a nice set of two numbers that are the same colour. By considering the possibilities for the colours of the numbers 1, 2, and 3, prove such a set always exists.

Suppose that we want to find a nice set of four numbers that are all the same colour. By considering the possibilities for the colours of the numbers from 1 to 27 inclusive, prove such a set always exists. [Hint: consider $\{ 1,2,3 \} , \{ 4,5,6 \} , \ldots , \{ 25,26,27 \}$.]

For $n = 3$, explain why the list $( 2,1,1 )$ is good, but the list $( 2,2,2 )$ is not good.

For $n = 3$, there are 16 good lists, so $G ( 3 ) = 16$. List all of them, starting with good lists with $t ( 1 ) = 1$, then good lists with $t ( 1 ) = 2$, and then good lists with $t ( 1 ) = 3$.

For $k = 1 , \ldots , n$, let $F ( n , k )$ be the number of good lists of length $n$ which result in team $n$ booking room $k$. Explain why $F ( n , k )$ is a multiple of $k$.

Describe the relationship between $G ( n )$ and $F ( n , 1 ) , F ( n , 2 ) , \ldots , F ( n , n )$.

Explain why $F ( 4,1 ) = G ( 3 )$ and $F ( 4,4 ) = 4 \times G ( 3 )$.

Find the values of $F ( 4,2 )$ and $F ( 4,3 )$. Explain your answers in each case. Hence find the value of $G ( 4 )$.

1. Let $f(x) = ax^6 - bx^4 + 3x - \sqrt{2}$, where $a, b$ are non-zero real numbers. Then the value of $f(5) - f(-5)$ is
(1) $-30$
(2) $0$
(3) $2\sqrt{2}$
(4) $30$
(5) Cannot be determined (depends on $a, b$)

4. A set of five cards each have a letter printed on their front and a number printed on their back, as follows:

	\begin{tabular}{ r r r r } Card A	Card B	Carc C	Card D
Fronts	A	B	Card E
\cline{2-4}		D	E

\end{tabular}
Backs [Figure]
Which one of the five cards (A, B, C, D or E) provides a counterexample to the following statement?
Every card that has a vowel on its front has an even number on its back.

8. Consider the following statement about the positive integer $n$ :
Statement (*): The sum of the four consecutive integers, the smallest of which is $n$, is a multiple of 6 .
Which one of the following is true?
A Statement () is true for all values of $n$.
B Statement () is true for all values of $n$ which are odd, but not for any other values of $n$.
C Statement (*) is true for all values of $n$ which are multiples of 3 , but not for any other values of $n$.
D Statement (*) is true for all values of $n$ which are multiples of 6 , but not for any other values of $n$.
E Statement (\textit{) is not true for any value of $n$.