Critical Sets and Unavoidable Sets of Strictly Concentric Magic Squares of Odd Order and Their Application to Prime Strictly Concentric Magic Squares of Order 5 †
Abstract
1. Introduction and Preliminaries
2. Completability of Partial SCMS with Non-Empty Centre Cell
- 1.
- If is a strong partial SCMS, then it is uniquely completable.
- 2.
- If is multiply completable, then it is a weak partial SCMS.
- 3.
- If is a weak partial SCMS, then it is either uniquely completable or multiply completable.
- 4.
- If is uniquely completable, then it is either a strong or a weak partial SCMS.
- 2.
- If a grid is multiply completable, then it cannot be a strong partial SCMS, so it is a weak partial SCMS.
- 3.
- This follows immediately from Lemma 1.
- 4.
- From conditions 1 and 3, both strong and weak partial SCMSs can be uniquely completable.
3. Strong Partial SCMSs and Critical Sets of Partial SCMSs
- 1.
- is the only SCMS of order n which has entry in position for all ;
- 2.
- No proper subset of satisfies (1).
4. Unavoidable Sets of SCMSs
- 1.
- The addition to of any triple , where , produces a partial SCMS, , which is uniquely completable; or
- 2.
- The addition to of any triple where , produces a partial SCMS, , which is either multiply completable to SCMS that are equivalent or possibly, for certain choices of triple , uniquely completable.
5. Classification of Unavoidable Sets of Order 5
5.1. Proper Unavoidable Sets of Minimum PSCMSs of Order 5
5.2. Improper Unavoidable Sets of Minimum PSCMSs of Order 5
- 1.
- Form 1, the minimum cardinality of the unavoidable set is 4.
- 2.
- Form 2, the minimum cardinality of the unavoidable set is 6.
- 3.
- Form 3, the minimum cardinality of the unavoidable set is 12.
- 4.
- Form 4, the minimum cardinality of the unavoidable set is 8.
- 5.
- Form 5, the minimum cardinality of the unavoidable set is 14.
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
NMS | Normal Magic Square |
CMS | Concentric Magic Square |
SCMS | Strictly Concentric Magic Square |
PSCMS | Prime Strictly Concentric Magic Square |
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Permutation | Order |
---|---|
Permute the border pairs in columns i, and , | |
noting that the values remain in their original row | |
Permute the border pairs in rows i, and , | |
noting that the values remain in their original column | |
Permute the value in with the value in | |
for all | 2 |
Permute the value in with the value in | |
for all | 2 |
Permute the value in with the value in | |
and the value in with the value in | |
for all where | 2 |
Type | List of Primes in the Border of Order 5 |
---|---|
1A | 11, 53, 71, 83, 101, 113, 191, 239, 263, 311, 389, 401, 419, 431, 449, 491 |
1B | 11, 53, 83, 101, 113, 149, 191, 239, 263, 311, 353, 389, 401, 419, 449, 491 |
1C | 11, 71, 83, 101, 113, 149, 191, 239, 263, 311, 353, 389, 401, 419, 431, 491 |
1D | 53, 71, 83, 101, 113, 149, 191, 239, 263, 311, 353, 389, 401, 419, 431, 449 |
1E | 11, 53, 71, 83, 101, 113, 149, 239, 263, 353, 389, 401, 419, 431, 449, 491 |
1F | 11, 53, 71, 83, 101, 149, 191, 239, 263, 311, 353, 401, 419, 431, 449, 491 |
2A | 11, 23, 41, 53, 101, 113, 149, 191, 311, 353, 389, 401, 449, 461, 479, 491 |
2B | 11, 23, 41, 53, 101, 113, 149, 233, 269, 353, 389, 401, 449, 461, 479, 491 |
2C | 11, 23, 41, 53, 101, 113, 191, 233, 269, 311, 389, 401, 449, 461, 479, 491 |
2D | 11, 23, 41, 101, 113, 149, 191, 233, 269, 311, 353, 389, 401, 461, 479, 491 |
2E | 11, 23, 41, 53, 101, 149, 191, 233, 269, 311, 353, 401, 449, 461, 479, 491 |
2F | 11, 23, 41, 53, 113, 149, 191, 233, 269, 311, 353, 389, 449, 461, 479, 491 |
2G | 11, 23, 53, 101, 113, 149, 191, 233, 269, 311, 353, 389, 401, 449, 479, 491 |
2H | 11, 41, 53, 101, 113, 149, 191, 233, 269, 311, 353, 389, 401, 449, 461, 491 |
2I | 23, 41, 53, 101, 113, 149, 191, 233, 269, 311, 353, 389, 401, 449, 461, 479 |
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Skelt, A.L.; Perkins, S.; Roach, P.A. Critical Sets and Unavoidable Sets of Strictly Concentric Magic Squares of Odd Order and Their Application to Prime Strictly Concentric Magic Squares of Order 5. Axioms 2025, 14, 607. https://doi.org/10.3390/axioms14080607
Skelt AL, Perkins S, Roach PA. Critical Sets and Unavoidable Sets of Strictly Concentric Magic Squares of Odd Order and Their Application to Prime Strictly Concentric Magic Squares of Order 5. Axioms. 2025; 14(8):607. https://doi.org/10.3390/axioms14080607
Chicago/Turabian StyleSkelt, Anna Louise, Stephanie Perkins, and Paul Alun Roach. 2025. "Critical Sets and Unavoidable Sets of Strictly Concentric Magic Squares of Odd Order and Their Application to Prime Strictly Concentric Magic Squares of Order 5" Axioms 14, no. 8: 607. https://doi.org/10.3390/axioms14080607
APA StyleSkelt, A. L., Perkins, S., & Roach, P. A. (2025). Critical Sets and Unavoidable Sets of Strictly Concentric Magic Squares of Odd Order and Their Application to Prime Strictly Concentric Magic Squares of Order 5. Axioms, 14(8), 607. https://doi.org/10.3390/axioms14080607