Congratulations to 5 of our math students on presenting their research at the Math Association of America (MAA) Seaway Section Meeting on April 14 at the College at Brockport. Gabriella Pesce ‘19, Samantha Todres ‘19, Molly Noel ‘20, Noelle Sullivan ‘19, and Daniel Akimchuk ‘19.
Molly Noel ’20 and Noelle Sullivan ‘19, Ithaca College
Extension of Kaprekar's Algorithm to Arbitrary Bases
Abstract: Kaprekar’s Algorithm was developed in 1949 by D.R. Kaprekar. One takes a 4 digit number in base 10 and creates two new numbers, one with the digits of the orginal number in descending order and the other with these digits in ascending order. Then one subtracts the number with digits in ascending order from the number with digits in descending order. The algorithm shows that repeating this process will always converge to 6174 no matter what the originating number. For 3 digit numbers in an arbitrary base n, we found that all Kaprekar numbers were of the form [α − 1][n − 1][n − α], where n is the base and α is the difference between the largest and smallest digit. We were also able to prove that when n is even, the algorithm converges to one fixed point, and we can compute this fixed point. We proved that for odd bases, the algorithm always converges to a cycle of two numbers, and we can compute these two numbers. Other work includes generalizing the Kaprekar algorithm for 4 digit numbers in an arbitrary base. We plan on looking at cycles for 4 digit numbers in arbitrary bases. We hope to find similar patterns like we did with the 3 digit cases. So far we know that we are not going to be able to generalize the 4 digit cases as even and odd like we did with the 3 digit case.
Daniel Akimchuk ‘19, Ithaca College
The Behavior of Rational Squares
Abstract: In his 2015 paper, On The Distribution of Rational Squares, Michael Weiss explores the behavior of rational squares - numbers that can be written in the form ( p q ) 2 for integers p and q - particularly which rational squares fall between certain consecutive integers. As a continuation of his research, we examined many patterns associated with these numbers, especially the connection between the size of the denominator and their distribution. Looking at the set of denominators that are part of a rational square that falls between two given consecutive integers n and n + 1, we were able to uncover properties regarding the distribution of the rational squares. In addition to this, we explored the minimum value σ of this set for given n, as depicted in the graph of σ(n) vs. n, which proved to be full of patterns. We prove that a square root function upper bounds the graph, all values of σ(n) will be greater than 2, and also formulas for σ(n) for certain patterns of n. We also look at the set Σc of denominators greater than the minimum that do not have a numerator allowing the square to fall between two certain consecutive integers. The size of this set proves to be very interesting, as there are certain patterns of integers that cause this set to be empty. Because a rational square with a given denominator, q, will produce an integer square once every q numerators, the set of squares is effectively “compressed.” We explore how this manner of “compression” determines which rational squares fall between which integers. Do all of the patterns that emerge in rational squares have counterparts when raising rational numbers to the third, fourth, or k-th power? Is there ever more than one rational number with denominator q that fall in between the same consecutive integers? If so, how can we determine how many/which numerators will satisfy this? Which consecutive integers can a rational square with denominator q fall between? We explore these questions to better understand the behavior of rational squares.
Gabriella Pesce ‘19 and Samantha Todres ‘19, Ithaca College
A Two-Piece Puzzle…How Hard Could It Be?
Abstract: A polyomino is defined as a plane geometric figure made up of one or more equal squares joined edge to edge; polyominoes appear in pop culture as playing pieces in the games of Dominoes and Tetris. We are studying polyominoes that can tile rectangles with two copies, where two polyominoes are considered the same if they differ by a rotation. Our results allow us to count the number of polyominoes that can tile a rectangle with dimensions 2N × 2M or (2N + 1) × 2M with only two copies, where M and N are positive integers. The process of counting these polyominoes involves matching different polyomino shapes with integer sequences. These integer sequences are constructed by counting the number of blocks that occur in each column of a polyomino. In many cases, a polyomino can be represented by more than one integer sequence. We will detail our method for discarding these excess shapes; this will then allow us to enumerate all polyominoes that can tile a rectangle of any specified dimension using two copies.
