The Institute of Mathematics and its Applications, 1998

Preface (Excerpts by Kathleen Ollerenshaw)

The enumeration of all 'most-perfect' squares of order *n* = 2^{r} (r > 1) that are pandiagonal magic squares with additional special properties and are defined in Chapter 1, was conjectured by me in 1987 and stated in the 25th Anniversary *IMA Bulletin* of March 1989. Formal publication of the proof was delayed because extensions of this result for other multiples of 4 were in sight. The full enumeration for *n* = 2^{r}*p*^{s} (*p* any prime > 2, r > 1, s ≥ 0) that forms the first part of this book was arrived at in 1989 and its existence mentioned in the *IMA Bulletin* article referred to above. However, the crux of the enumeration (and construction) that is contained in Chapter 4 and on which the result for all most-perfect squares depends had been based merely on intuition and a strict adherence to symmetries and pattern; there had been no attempt to find a rigorous proof. When a serious effort was made to provide proof, the argument became increasingly complex and involved numerous diversions that had their own interest. The full proof thus became unsuitable for publication as an article in a recognized journal and better suited to publication as a book. From the beginning of the search for proof, I have always had the comfort and certainty of an assuredly correct answer. The algebraic 'discoveries' emerging during the course of the work – new to me if not to others – have been a continuing source of elation.

My inclination in trying to solve problems of this kind is to follow patterns of behaviour and work from the particular to the general – no basis whatever for dealing with functions that are to extend to infinity, particularly when, as here, the resulting equations are not amenable to the classic method of proof by induction. Instead, the proof of the construction and initial formulae for the enumeration of squares has been arrived at through logical argument At every stage, the central issue has been that of binomial coefficients, typically 'in how many different ways can these particular choices be made form those several possibilities?'. The final formula for the enumeration itself has been deduced, in Chapter 6, by applying the appropriate algebraic identities involving series of binomial coefficients found (usually in different format from my own) in the textbook *Concrete Mathematics* (Graham, Knuth and Patashnik, 1989).

**Contents**

**Pandiagonal magic squares**

1.1 Introduction

1.2 Brief history of magic squares

1.3 Pandiagonal magic squares

1.4 Most-perfect squares

1.5 In search of a method for construction most-perfect squares

1.6 Overview

**Most-perfect and reversible squares**

2.1 The need for a simpler square

2.2 Most-perfect squares

2.3 Reversible squares

2.4 Transformations of reversible squares that lead to other reversible squares

2.5 Principal reversible squares

2.6 Summary

**Mappling between most-perfect and doubly-even reversible squares**

3.1 Transforming a doubly-even reversible square into a most-perfect square

3.2 Every most-perfect square can be reached from a reversible square

3.3 Summary

**The construction of principal reversible squares**

4.1 Principal reversible squares can be constructed from blocks

4.2 A principal reversible square must be constructed from similar blocks

4.3 The smallest corner block, B_{0}(*n*_{1},*f*_{1})

4.4 Constructing successively larger corner blocks from smaller blocks

4.5 Completing the square

4.6 The square is a principal reversible square

**Enumerating the different configurations of largest corner blocks**

5.1 The objective

5.2 The number of ways of choosing ν progressive factors of*f*and of*n*

5.3 Enumerating the different configurations*F*(_{n}*f*) for B_{0}(*n*,*f*)

5.4 Summary

**Enumerating reversible squares when***n*= 2^{r}*p*^{s}

6.1 Ways of selecting the largest corner block

6.2 Summing over progressive factors

6.3 Rearranging binomial coefficients

6.4 Finding the closed form for the summation over*j*

6.5 Recombining binomial coefficients

6.6 Enumeration for powers of 2

6.7 The total number of reversible squares

**Enumerating all doubly-even reversible squares**

7.1 Enumerating principal reversible squares when*n*is doubly-even

7.2 The number of ways of choosing progressive factors

7.3 Enumeration for small values of*n*

7.4 Summary

**Conclusion**

8.1 On methods of construction

8.2 Construction by using Latin auxiliary squares

8.3 Construction by using paths

8.4 Construction by using the chess knight's path and mixed auxiliary squares

8.5 In conclusion

8.6 Summary

**A personal perspective**

**Glossary**

**Appendices**

- Properties of binomial coefficients
- Proofs of some properties of most-perfect and reversible squares
- Methods for constructing pandiagonal squares
- Construction of most-perfect squares from reversible squares
- Complete list of principal reversible squares of order 12