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23

11

Article 11.8.8

Journal of Integer Sequences, Vol. 14 (2011),

2

3

6

1

47

The Inverse Football Pool Problem

David Brink

Institut for Matematiske Fag

Københavns Universitet

Universitetsparken 5

DK-2100

Denmark

brink@math.ku.dk

Abstract

The minimal number of spheres (without “interior”) of radius n required to cover

the ﬁnite set {0, . . . , q−1}

n

equipped with the Hamming distance is denoted by T (n, q).

The only hitherto known values of T (n, q) are T (n, 3) for n = 1, . . . , 6. These were all

given in the 1950s in the Finnish football po ol magazine Veikkaaja along with upper

and lower bounds for T (n, 3) for n ≥ 7. Recently,

¨

Osterg˚ard and Riihonen found

improved upper bounds for T (n, 3) for n = 9, 10, 11, 13 using tabu search. In the

present paper, a new method to determine T (n, q) is presented. This method is used

to ﬁnd the next two values of T (n, 3) as well as six non-trivial values of T (n, q) with

q > 3. It is also shown that, modulo equivalence, there is only one minimal covering of

{0, 1, 2}

n

for each n ≤ 7, thereby proving a conjecture of

¨

Osterg˚ard and Riihonen. For

reasons discussed in the paper, it is proposed to denote the problem of determining

the values of T (n, 3) as the inverse football pool problem.

1 Introduction

Consider the ﬁnite set Q = {0, . . . , q − 1}

n

equipped with the Hamming distance

d(x, y) = |{i : x

i

6= y

i

}|

for x = (x

1

, . . . , x

n

) and y = (y

1

, . . . , y

n

) in Q. Let

B(x

0

, r) = {x ∈ Q : d(x, x

0

) ≤ r}, S (x

0

, r) = {x ∈ Q : d(x, x

0

) = r}

be the ball and sphere, respectively, with centre x

0

and radius r.

1

1

Note that the word “sphere” is often used in the literature for what is here called a “ball.”

1

These simple concepts give rise to a plethora of combinatorial problems. The most famous

is the football pool problem which asks for the minimal number K

3

(n, 1) of balls of radius 1

required to cover {0, 1, 2}

n

. In the football pools, one bets simultaneously on the outcomes of

a number of football games, typically n = 12 or 13. Each game has th ree possible outcomes:

win, draw or loss. A bet can thus be thought of as a point in {0, 1, 2}

n

, and K

3

(n, 1) is equal

to the minimal number of bets on n games required to guarantee at least one bet with at

most one wrong outcome.

The complexity of such problems increases drastically with n. Indeed, when n is not of the

form (3

m

− 1)/2 and the existence of a perfect Hamming code shows K

3

(n, 1) = 3

n

/(2n + 1),

then K

3

(n, 1) is only known for n = 2, 3, 5, cf. Sloane’s A004044 [

5]. See also [2] for an

interesting discussion of this and related problems.

In the present paper, we study coverings of Q with spheres of radius n. S uch spheres are

n-dimensional grid graphs, i.e., they are of the form

Q

n

i=1

A

i

where A

i

⊂ {0, . . . , q − 1} with

|A

i

| = q − 1. The function deﬁned below therefore equals the special case T (n, q, q − 1) of

the function T (n, q, p) introduced by

¨

Osterg˚ard and Riihonen [

3].

Deﬁnition 1. Let T (n, q) be the minimal number of spheres of radius n required to cover Q.

We shall be particularly concerned with the problem of determining the values of T (n, 3),

Sloane’s A086676. We propose to call this the inverse football pool problem since it means

asking for the minimal number of bets needed to guarantee that at least one is totally wrong.

The inverse football pool problem was ﬁrst considered in the Finnish football pool mag-

azine Veikkaaja in the 1950s.

2

The principal goal then was to determine T(12, 3), a value

which remains unknown today. The only hitherto known values of T (n, q) are T(1, 3) = 2,

T (2, 3) = 3, T(3, 3) = 5, T (4, 3) = 8, T (5, 3) = 12, and T (6, 3) = 18. The history of these

results is discussed below.

The results in Veikkaaja on the inverse football pool problem seem to have been unknown

to the mathematical community until they were discovered by

¨

Osterg˚ard and Riihonen in

2002 and presented in [

3] and [4]. All references to Veikkaaja here come from these two

papers.

2 General bounds

First we state some easy or well-known bounds. If a sphere covering of {0, . . . , q − 1}

n

is

given, a covering of {0, . . . , q − 1}

n−1

is obtained by discarding the last coordinate as well

as all spheres whose centres have any given value in that coordinate, cf. Veikkaaja no. 23,

1956, and [

3, Theorem 3.3]. Hence

T (n, q) ≥

q

q − 1

· T (n − 1, q). (1)

2

As was the ordinary football pool problem in the previous decade. Indeed, Veikkaaja holds a place of

the ﬁrst rank in the history of combinatorics. It was here that Juhani Virtakallio gave the ternary Golay

code (a perfect covering of {0, 1, 2}

11

with 729 balls of radius 2) in 1947, two years before its rediscovery by

Marcel Golay.

2

In particular, it follows inductively from T (1, q) = 2 that T (n, q) ≥ n + 1. If n < q, the

spheres with centres (0, . . . , 0), . . . , (n, . . . , n) cover Q. Thus

T (n, q) = n + 1 for n < q. (2)

Taking direct product gives

T (n + n

′

, q) ≤ T (n, q) · T (n

′

, q). (3)

Finally, an exceedingly elegant and ingenious construction for q = 3, described in Veikkaaja

no. 15, 1960, and [

3, Corollary 3.5], gives the bound

T (n + n

′

− 2, 3) ≤

1

3

· T (n, 3) · T (n

′

, 3) (4)

(and sometimes a little more, cf. [

3, Theorem 3.4]) which is considerably better than (3).

Deﬁnition 2. (a) We represent a covering of {0, . . . , q − 1}

n

with m spheres S (x

1

, n), . . . ,

S (x

m

, n) by the n × m covering matrix A whose columns are the centres x

1

, . . . , x

m

.

(b) Two covering matrices of the same size (and the sphere coverings they represent) are

called equivalent if one can be transformed into the other by permuting rows and columns

and by renaming the symbols in any given row.

3

(c) We call a covering matrix canonical if it is minimal among all equivalent matrices

with resp ect to the lexicographic ordering (reading ﬁrst row 1 from left to right, then row 2

from left to right, etc.).

The deﬁnition of equivalence is standard. If n < q, for example, the matrix consisting of

n identical rows (0, 1, . . . , n) is the unique canonical n×(n+1) covering matrix, cf. (

2). Note

that there is a 1–1 correspondence between equivalence classes of coverings and canonical

covering matrices. The following lemma is a generalization of (

1).

Lemma 3. Suppose A is an n × m covering matrix. Let there be given b

i

∈ {0, . . . , q − 1}

for all i in some proper subset I ⊂ {1, . . . , n}. Then the number of columns (a

1

, . . . , a

n

)

t

of

A with a

i

6= b

i

for all i ∈ I is at least T (n − |I|, q).

Proof. For each i ∈ I, delete all columns (a

1

, . . . , a

n

)

t

of A having a

i

= b

i

. Then delete row i

for each i ∈ I. The resulting matrix A

′

is a covering matrix for {0, . . . , q − 1}

n−|I|

and hence

has at least T (n − |I|, q) columns.

3 Main results

Theorem 4. We have the following two values of T (n, 3), i.e., the minimal number of

spheres of radius n needed to cover {0, 1, 2}

n

: T (7, 3) = 29 and T (8, 3) = 44.

3

By “renaming the symbols,” we mean that given any row a = (a

1

, . . . , a

m

) and given any permutation

σ of 0, . . . , q − 1, we may replace a by (σ(a

1

), . . . , σ(a

m

)).

3

Proof. The lower bounds T (7, 3) ≥ 27 and T (8, 3) ≥ 41 follow from (1) and (2). The upper

bounds T (7, 3) ≤ 29 and T (8, 3) ≤ 44 are known, cf. [

3] and the discussion below. After

having shown T (7, 3) = 29, it will follow that T (8, 3) = 44 by (

1). The rest of the proof

consists of three parts. F irst we describe in general our method for ﬁnding canonical covering

matrices. Then we illustrate this method by proving in detail the bound T (3, 7) > 27. Finally

we discuss how the method can be used to prove T (3, 7) > 28.

Suppose A is a canonical n × m covering matrix. Then, by Deﬁnition 2 and Lemma 3,

the following two conditions hold for each i = 1, . . . , n:

(I) The i × m submatrix A

i

consisting of the ﬁrst i rows of A is canonical.

(II) Given b

1

, . . . , b

i

∈ {0, . . . , q − 1}, there are at least T (n − i, q) columns (a

1

, . . . , a

i

)

t

of

A

i

satisfying a

j

6= b

j

for all j = 1, . . . , i.

In order to ﬁnd all canonical n × m matrices, we proceed as follows: First ﬁnd the set R

1

of

all rows r

1

such that the 1×m matrix A

1

consisting only of r

1

satisﬁes (I) and (II). Then, for

each r

1

∈ R

1

, ﬁnd the set R

2

(r

1

) of all rows r

2

such that the 2 × m matrix A

2

consisting of r

1

and r

2

satisﬁes (I) and (II). Next, for each r

1

∈ R

1

and r

2

∈ R

2

(r

1

), ﬁnd the set R

3

(r

1

, r

2

),

and so forth. Every time a non-empty set R

n

(r

1

, . . . , r

n−1

) is obtained in this way, each row

r

n

in that set gives rise to a canonical n × m covering matrix, namely the matrix consisting

of r

1

, . . . , r

n

.

To illustrate the method just described, we prove in detail the bound T (7, 3) > 27.

Suppose indirectly that a covering of {0, 1, 2}

7

is given as a canonical 7 × 27 covering matrix

A. First we ﬁnd R

1

. By condition (I), every r

1

∈ R

1

is of the form

r

1

= (

l

0

times

z

}| {

0, . . . , 0,

l

1

times

z }| {

1, . . . , 1,

l

2

times

z }| {

2, . . . , 2)

with

l

0

+ l

1

+ l

2

= 27. (5)

By specifying b

1

∈ {0, 1, 2} and using T (6, 3) = 18, condition (II) gives the following three

linear inequalities:

l

1

+ l

2

≥ 18, l

0

+ l

2

≥ 18, l

0

+ l

1

≥ 18.

(6)

It follows immediate from (

5) and (6) that l

0

= l

1

= l

2

= 9 and hence

r

1

= (0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2).

Then we ﬁnd R

2

(r

1

). By condition (I), every r

2

∈ R

2

(r

1

) is of the form

r

2

= (

l

0

times

z

}| {

0, . . . , 0,

l

1

times

z

}| {

1, . . . , 1,

l

2

times

z

}| {

2, . . . , 2,

l

′

0

times

z

}| {

0, . . . , 0,

l

′

1

times

z

}| {

1, . . . , 1,

l

′

2

times

z

}| {

2, . . . , 2,

l

′′

0

times

z

}| {

0, . . . , 0,

l

′′

1

times

z

}| {

1, . . . , 1,

l

′′

2

times

z

}| {

2, . . . , 2)

with

l

0

+ l

1

+ l

2

= l

′

0

+ l

′

1

+ l

′

2

= l

′′

0

+ l

′′

1

+ l

′′

2

= 9. (7)

4

By specifying b

1

, b

2

∈ {0, 1, 2} and using T (5, 3) = 12, condition (II) now gives the following

nine linear inequalities:

l

′

1

+ l

′

2

+ l

′′

1

+ l

′′

2

≥ 12, l

′

0

+ l

′

2

+ l

′′

0

+ l

′′

2

≥ 12, l

′

0

+ l

′

1

+ l

′′

0

+ l

′′

1

≥ 12,

l

1

+ l

2

+ l

′′

1

+ l

′′

2

≥ 12, l

0

+ l

2

+ l

′′

0

+ l

′′

2

≥ 12, l

0

+ l

1

+ l

′′

0

+ l

′′

1

≥ 12,

l

1

+ l

2

+ l

′

1

+ l

′

2

≥ 12, l

0

+ l

2

+ l

′

0

+ l

′

2

≥ 12, l

0

+ l

1

+ l

′

0

+ l

′

1

≥ 12.

(8)

It is not diﬃcult to show that (

7) and (8) imply l

0

= l

1

= l

2

= l

′

0

= l

′

1

= l

′

2

= l

′′

0

= l

′′

1

= l

′′

2

= 3

and consequently

r

2

= (0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 1, 1, 1, 2, 2, 2).

Similarly, in order to determine the elements r

3

∈ R

3

(r

1

, r

2

), we get from (I), (II), and

T (4, 3) = 8 a system of 27 linear inequalities in 27 unknowns which together imply

r

3

= (0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2).

The determination of R

4

(r

1

, r

2

, r

3

) is somewhat more complicated. It is possible by hand to

proceed as above and show that this set consists only of

r

4

= (0, 1, 2, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 2, 0).

It is then relatively easy to show that R

5

(r

1

, r

2

, r

3

, r

4

) is empty and that in consequence

no 7 × 27 covering matrix of {0, 1, 2}

7

exists. However, it is also possible at this point to

make a short-cut. The determination of r

1

, r

2

, and r

3

shows that every possible column

vector (b

1

, b

2

, b

3

)

t

with b

i

∈ {0, 1, 2} occurs exactly λ = 1 times in the 3 × 27 matrix A

3

. By

symmetry, the same holds for any 3 × 27 submatrix of A formed by t = 3 arbitrary rows.

This means that the 7 × 27 matrix A is an orthogonal array with q = 3 levels, index λ = 1,

and strength t = 3. But by a result of Bush [

1], a such matrix can have at most t + 1 = 4

rows, thus disproving the existence of A.

The method used to prove T (7, 3) > 28 is exactly the same, but now the row sets R

i

for

the hypothetical 7 × 28 covering matrix become too large to do the demonstration by hand.

For example, R

1

consists of the two rows

r

1

= (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2),

r

′

1

= (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2),

and R

2

(r

1

) consists of

r

2

= (0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 0, 0, 1, 1, 2, 2, 2, 2),

r

′

2

= (0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 1, 1, 1, 2, 2, 2),

while R

2

(r

′

1

) consists of

r

2

= (0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 1, 1, 1, 1, 2, 2, 0, 0, 0, 1, 1, 2, 2, 2, 2),

r

′

2

= (0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 1, 1, 1, 2, 2, 2).

Instead, the method can be implemented on a computer to show that no such matrix exists.

This computation took 28 seconds on the author’s laptop.

5

n T (n, 3)

1 2

2

3

3

5

4

8

5

12

6

18

7

29

8

44

9

66 − 68

10

99 − 104

11

149 − 172

12

224 − 264

13

336 − 408

Table 1: Old and new results on T (n, 3).

Table 1 collects old and new results on T (n, 3). All values for n ≤ 6 as well as the bounds

T (7, 3) ≤ 29 and T (8, 3) ≤ 44 go back to Veikkaaja.

4

All lower bounds for n ≥ 9 come

from T(7, 3) = 29 combined with (

1). Note that T (7, 3) = 29 is the ﬁrst counter-example

to the natural conjecture T (n, 3) = ⌈

3

2

· T (n − 1, 3)⌉. The upper bounds for n = 9, 10 were

found in [

3] using so-called t abu search. The upper bounds for n = 11, 12, 13 follow from the

upper bounds for smaller n using (

4). A covering proving T (12, 3) ≤ 264 appears explicitly

in Veikkaaja no. 52, 1960. As related in [

4], a writer in fact claims in Veikkaaja no. 15, 1960,

to b e in possession of a covering proving T (12, 3) ≤ 242, but states—not unlike Pierre de

Fermat three centuries earlier—that it is to o long to be printed in the magazine!

The case n = 7 of the following theorem was conjectured in [

3].

Theorem 5. Modulo equivalence, there is only one minimal cov ering of {0, 1, 2}

n

with

spheres of radius n for each n = 1, . . . , 7.

Proof. The statement can b e proved by hand for n ≤ 6. We exemplify this by sketching the

case n = 6. By the same arguments as in the proof of Theorem

4, the ﬁrst three rows of a

canonical 6 × 18 covering matrix A must be

r

1

= (0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2),

r

2

= (0, 0, 1, 1, 2, 2, 0, 0, 1, 1, 2, 2, 0, 0, 1, 1, 2, 2),

r

3

= (0, 1, 0, 2, 1, 2, 0, 2, 1, 2, 0, 1, 1, 2, 0, 1, 0, 2).

Now there are three candidates for row 4:

r

4

= (0, 1, 1, 2, 2, 0, 2, 1, 2, 0, 1, 0, 0, 2, 0, 1, 2, 1),

r

′

4

= (0, 1, 2, 0, 2, 1, 2, 0, 1, 2, 1, 0, 2, 1, 1, 0, 0, 2),

r

′′

4

= (0, 1, 2, 1, 0, 2, 1, 2, 2, 0, 0, 1, 2, 0, 1, 0, 2, 1).

But also rows 5 and 6 must be chosen among r

4

, r

′

4

, and r

′′

4

. One can check directly that the

only way to get a covering is to take one of each. Since the rows of a canonical matrix must

come in increasing order, rows 4–6 of A must be r

4

, r

′

4

, and r

′′

4

in that order.

The case n = 7 is too complicated to settle by hand, but an exhaustive computer search

following the same principles as described in the proof of Theorem 4 ﬁnds only one canonical

matrix (namely the one given at the end of this paper).

4

It remains somewhat of a mystery, though, how the coverings proving these results were found.

6

We do not know if the covering of {0, 1, 2}

8

with 44 spheres is unique.

Next we consider values of T (n, q) with q > 3 (the case q = 2 being trivial).

Theorem 6. We have the following values of T (n, q), i.e., the minimal number of spheres

of radius n needed to cover {0, . . . , q − 1}

n

:

n

T (n, 4) T (n, 5) T (n, 6) T (n, 7)

1 2 2 2 2

2

3 3 3 3

3

4 4 4 4

4

7 5 5 5

5

10 8 6 6

6

14 − 16 11 10 7

7

19 − 28 14 − 20 12 − 18 11

The values typed in boldface are the non-trivial ones, i.e., the ones not following from equation

(2).

Proof. The theorem is shown by a computer search following the same principles as in the

proof of Theorem

4. All results were found within 24 hours of computer time, some of them

within minutes or seconds. A few non-trivial partial results can be proved by hand. For

example, one can see T (4, 4) > 6 by showing that rows 1–3 of a canonical 4 × 6 covering

matrix must necessarily be (0, 0, 1, 1, 2, 3), (0, 1, 0, 1, 2, 3), (0, 1, 1 ,0, 2, 3), but that

this leaves no possibilities for row 4. All lower and upper bounds in the undecided cases are

straightforward consequences of (

1) and (3).

It appears that minimal coverings are more abundant when q is composite.

4 Appendix: Lexicographically minimal covering ma-

trices

Finally, we present for each non-trivial value of T (n, q) given in Theorem

6 as well as for

q = 3 and n = 3, . . . , 7 the unique lexicographically minimal n × T (n, q) covering matrix.

T (3, 3) = 5: T (4, 3) = 8: T (5, 3) = 12:

(0, 0, 1, 1, 2) (0, 0, 0, 1, 1, 1, 2, 2) (0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2)

(0, 1, 0, 1, 2) (0, 0, 1, 0, 1, 1, 2, 2) (0, 0, 1, 2, 0, 1, 1, 2, 0, 1, 2, 2)

(0, 1, 1, 0, 2) (0, 1, 2, 2, 0, 1, 0, 1) (0, 1, 0, 2, 1, 0, 1, 2, 2, 2, 0, 1)

(0, 1, 2, 2, 0, 1, 1, 0) (0, 1, 2, 1, 2, 1, 0, 2, 0, 2, 1, 0)

(0, 1, 2, 2, 2, 1, 0, 1, 2, 0, 0, 1)

7

T (6, 3) = 18:

(0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2)

(0, 0, 1, 1, 2, 2, 0, 0, 1, 1, 2, 2, 0, 0, 1, 1, 2, 2)

(0, 1, 0, 2, 1, 2, 0, 2, 1, 2, 0, 1, 1, 2, 0, 1, 0, 2)

(0, 1, 1, 2, 2, 0, 2, 1, 2, 0, 1, 0, 0, 2, 0, 1, 2, 1)

(0, 1, 2, 0, 2, 1, 2, 0, 1, 2, 1, 0, 2, 1, 1, 0, 0, 2)

(0, 1, 2, 1, 0, 2, 1, 2, 2, 0, 0, 1, 2, 0, 1, 0, 2, 1)

T (7, 3) = 29:

(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2)

(0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 0, 0, 1, 1, 2, 2, 2, 2)

(0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 2, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 2, 2, 2, 2, 0, 0, 1, 1)

(0, 1, 0, 2, 0, 2, 1, 2, 0, 1, 2, 1, 2, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 1, 2, 0, 1, 1, 2)

(0, 1, 1, 2, 2, 1, 2, 0, 1, 0, 2, 2, 0, 2, 1, 1, 2, 0, 1, 2, 1, 1, 0, 2, 0, 0, 1, 0, 2)

(0, 1, 2, 0, 2, 0, 1, 2, 0, 1, 2, 1, 2, 2, 0, 2, 0, 0, 1, 0, 2, 1, 0, 2, 1, 1, 0, 2, 1)

(0, 1, 2, 1, 1, 2, 2, 0, 1, 0, 2, 2, 0, 1, 2, 2, 1, 0, 1, 2, 1, 0, 1, 0, 2, 1, 0, 2, 0)

T (4, 4) = 7: T (5, 4) = 10: T (5, 5) = 8:

(0, 0, 0, 1, 1, 1, 2) (0, 0, 0, 1, 1, 1, 2, 2, 2, 3) (0, 0, 0, 1, 1, 1, 2, 3)

(0, 1, 2, 0, 1, 2, 3) (0, 0, 0, 1, 1, 1, 2, 2, 2, 3) (0, 1, 2, 0, 1, 2, 3, 4)

(0, 1, 2, 0, 1, 2, 3) (0, 1, 2, 0, 1, 2, 0, 1, 2, 3) (0, 1, 2, 0, 1, 2, 3, 4)

(0, 1, 2, 1, 2, 0, 3) (0, 1, 2, 0, 1, 2, 0, 1, 2, 3) (0, 1, 2, 1, 2, 0, 3, 4)

(0, 1, 2, 1, 2, 0, 2, 0, 1, 3) (0, 1, 2, 1, 2, 0, 3, 4)

T (6, 5) = 11: T (6, 6) = 10:

(0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4) (0, 0, 0, 0, 1, 1, 1, 1, 2, 3)

(0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4) (0, 1, 2, 3, 0, 1, 2, 3, 4, 5)

(0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 4) (0, 1, 2, 3, 0, 1, 2, 3, 4, 5)

(0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 4) (0, 1, 2, 3, 0, 1, 2, 3, 4, 5)

(0, 1, 2, 1, 2, 0, 2, 0, 1, 3, 4) (0, 1, 2, 3, 1, 0, 3, 2, 4, 5)

(0, 1, 2, 1, 2, 0, 2, 0, 1, 3, 4) (0, 1, 2, 3, 2, 3, 0, 1, 4, 5)

T (7, 7) = 11:

(0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 4)

(0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 6)

(0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 6)

(0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 6)

(0, 1, 2, 3, 1, 2, 3, 0, 4, 5, 6)

(0, 1, 2, 3, 1, 2, 3, 0, 4, 5, 6)

(0, 1, 2, 3, 1, 2, 3, 0, 4, 5, 6)

References

[1] K. A. Bush, Orthogonal arrays of index unity, Ann. Math. Statistics 23 (1952), 426–434.

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[2] H. H¨am¨al¨ainen, I. Honkala, S. Litsyn, and P. R. J.

¨

Osterg˚ard, Football pools—a game

for mathematicians, Amer. Math. Monthly 102 (1995), 579–588.

[3] P. R. J.

¨

Osterg˚ard and T. Riihonen, A covering problem for tori, Ann. Comb. 7 (2003),

357–363.

[4] T. Riihonen, How to gamble 0 correct in football pools, Helsinki University of Technol-

ogy, 2002. Available at

http://users.tkk.fi/priihone/tuotokset.html.

[5] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences. Published electroni-

cally at

http://oeis.org/.

2010 Mathematics Subject Classiﬁcation: Primary 05B40; Secondary 11H31, 52C17.

Keywords: Minimal covering codes, football pool problem.

Concerned with sequences A004044 and A086676.

Received January 14 2011; revised versions received February 19 2011; June 28 2011; Septem-

ber 5 2011. Published in Journal of Integer Sequences, October 16 2011.

Return to Journal of Integer Sequences home page.

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