Finding combinations of four-operations with Type-2 tree structure

Abstract

Combination problems are one of the most important issues of probability theory. The four-operations combination problem underlies the basis of some competition programs broadcasted in many national channels. In these competition programs, the competitors are expected to reach the target number by using six numbers and four basic arithmetic operators. The numbers are used at most once, the operators can be used any desired number to reach the target number. In this problem, all four-operations combinations include the operation blocks consisting of two numbers and an operator. Therefore, the four-operations combination problem is solved by developing a "Type-2 Tree Structure" which is a new approach to accurately model the operation blocks. The performance of the proposed method for the four-operations combination problem is examined by a simulation study. Also, the statistics from experimental results are given in this study.

Keywords

• Combination
• Four-Operations
• Type-2 tree
• Operation block

References

1. [1] Colton S. “Countdown numbers game: solved, analysed, extended”. Proceedings of the AISB Symposium on AI and Games, Canterbury, London, 1 April 2014.
2. [2] Defays D. L'esprit en Friche: les Foisonnements de l'intelligence Artificielle. Liege, Belgium, Pierre Mardaga, 1988.
3. [3] Defays D. Numbo: A Study in Cognition and Recognition. Editor: Hofstadter D. Fluid Concepts and Creative Analogies, 131-154, Basic Books Inc, 1995.
4. [4] Hutton G. “The countdown problem”. Journal of Functional Programming, 12(06), 609-616, 2002.
5. [5] Alliot JM. “The (Final) Countdown”. https://arxiv.org/abs/1502.05450 (03.06.2020).
6. [6] Fischler MA, Bolles RC. “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography”. Communications of the ACM, 24(6), 381-395, 1981.
7. [7] Montgomery DC, Runger GC. Applied Statistics and Probability for Engineers. 5th ed. Jefferson City, USA, John Wiley & Sons, 2011.
8. [8] Hines WW, Montgomery DC, Goldsman DM, Borror CM. Probability and Statistics in Engineering. Danvers, Massachusetts, USA, John Wiley & Sons, 2008.

9. [9] Code D. Probability: Mastering Permutations and Combinations. 2nd ed. California, USA, CreateSpace Independent Publishing Platform, 2017.
10. [10] Crawshaw J, Chambers J. A Concise Course in Advanced Level Statistics: With Worked Examples. 6th ed. London, UK, Oxford University Press, 2015.
11. [11] Tier R. Probability with Permutations and Combinations: A Deeper and More Thorough Look at the Fundamental Equations. California, USA, CreateSpace Independent Publishing Platform, 2017.
12. [12] Nicolaides A. Pure Mathematics Series: 10. Combinations, Permutations, Probabilities. London, UK, PASS Publications, 1994.
13. [13] Puntambekar A. Analysis and design of algorithms. 1st ed. Pune, India, Technical Publications, 2008.
14. [14] Mueller J, Massaron L. Algorithms for Dummies. Hoboken, USA, John Wiley & Sons, 2017.
15. [15] Goodrich MT, Tamassia R, Goldwasser, MH. Data Structures and Algorithms in Java. 6th ed. Hoboken, USA, John Wiley & Sons, 2014.
16. [16] Saha S, Shukla S. Advanced Data Structures: Theory and Applications. Boca Raton, USA, CRC Press: Taylor & Francis, 2019.