Three integers a, b and c satisfy both 0 ≤ a ≤ b ≤ c and
abc + ab + ac + bc + a + b + c = 2023
What is the smallest possible value for a + b + c?
Solution to Toy Box – WQ2024 #1
Let A, B, and C be the centers of the smaller circles, let E be the point of tangency of circles B and C, and let D be the center of the large circle, as indicated in the figure below.
Then triangle ABC is an equilateral triangle with side lengths 2r, and triangle CED is a right triangle with length of CE equal to r. Further, because triangle ABC is equilateral, angle Θ is half of angle ACB, or π/6 . If R is the radius of the large circle, then clearly
Now, 
so that
and therefore, 
Hence,
MAT Community Lunch for Women and Gender Queer Students and Faculty on 3/1
RSVP link: https://forms.gle/Bmg18bx3cBjt1uLb7
DePaul Math Club
The DePaul Math Club meets Fridays at 12:00 in Arts and Letters 410. It’s a great time filled with pizza, beverages, great people, a relaxed environment, and some fascinating mathematics—math that’s both niche and underdiscussed and the hottest topics leading the math world today. We are usually joined by student researchers, graduate students, or professors—some from DePaul and some from other institutions—to share the math they work with. No math background required. We hope that you will join us for one of our upcoming meetings:
- Friday, 2/23* – Talk by Dr. Desale Habtzghi
- Friday, 3/1 – Talk by Lucas Lam
- Friday, 3/8* – Talk by former DMC President
After the Math Club Meetings on 2/23 and 3/8, there will also be MAT Community Tea & Coffee Hours back in the Math Dept. on the 5th floor of SAC.
“Journeys of Black Mathematicians: Forging Resilience” on F 2/23
In honor of Black History Month, the Department of Mathematical Sciences invites all members of the DePaul community to join us for a screening of the new documentary “Journeys of Black Mathematicians – Forging Resilience.” Snacks and refreshments will be provided.
When: Friday, February 23, 3:00-4:30 pm
Where: McGowan South 104
Details: Forging Resilience, a documentary project by George Csicsery, traces the evolution of a culture of Black scholars, scientists and educators. The film follows the stories of prominent pioneers, showing how the challenges they faced, and their triumphs are reflected in the experiences of today’s working Black mathematicians. Their mathematical descendants in turn are contemporary college students, and K-12 children across the U.S. who are learning that they belong in mathematics and STEM. Check out the film’s website here.
Bonus: A recent DePaul graduate is featured in the film!
MAT Community Tea & Coffee Hours on 2/23 & 3/8
Throughout the year, the MAT Department hosts a number of informal events meant to help build and sustain our mathematical sciences community here at DePaul. These events provide an opportunity to meet with fellow MAT students, at both undergraduate and graduate levels, as well as MAT faculty members in a relaxed setting.
We will have two more teas this quarter:
- Friday, 2/23 at 1:30-2:45 pm
- Friday, 3/8 at 1:30-2:45 pm
These community hours will take place in the MAT Department office suite (on the 5th floor of SAC). Light refreshments, such as coffee, tea, and cookies, will be served. We hope that you can join us.
From the Mathematical Toy Box… (WQ2024)
In the diagram below, the radius of each of the smaller circles is r. What is the radius of the larger circle?
Solution to Toy Box – Fall 2023
Problem: If all possible (distinguishably different) arrangements of the letters A, C, C, L, L, S, U, and U are put in alphabetical order, in what position is the word “CALCULUS”?
Solution: The total number of distinguishably different arrangements of the letters A, C, C, L, L, S, U, and U is given by
To find the appearance of “CALCULUS,” we first, we need to “clear” all the arrangements that begin with A, of which there are
Those arrangements that begin with “CAC” will appear next on the list and need to be cleared as well. There are
of these, the number of distinguishable arrangements of the remaining letters L, L, S, U, and U.
The next several entries on the list will begin with “CALC,” followed by the letters L, S, U, and U. We can first clear the arrangements that start with “CALCL,” numbering
and then those that begin “CALCS,” which also number
Then the next arrangement on the list is “CALCULSU,” followed immediately by “CALCULUS.” Hence, “CALCULUS” appears in position number
630 + 30 + 3 + 3 + 1 + 1 = 668.
MAT Community Tea & Coffee Hour on F 2/2
Throughout the year, the Math Department hosts a number of informal events meant to help build and sustain our mathematical sciences community here at DePaul. These events provide an opportunity to meet with fellow MAT students, at both undergraduate and graduate levels, as well as MAT faculty members in a relaxed setting.
Our next event will take place on Friday, 2/2 at 10:30-11:45 am in the MAT Department office suite (on the 5th floor of SAC). Light refreshments, such as coffee, tea, and cookies, will be served. We hope that you can join us.
MAT Community Tea & Coffee Hour on 1/19
Throughout the year, the Math Department hosts a number of informal events meant to help build and sustain our mathematical sciences community here at DePaul. These events provide an opportunity to meet with fellow MAT students, at both undergraduate and graduate levels, as well as MAT faculty members in a relaxed setting.
Our next event will take place this Friday, 1/19 at 1:30-2:45 pm in the MAT Department office suite (on the 5th floor of SAC). Light refreshments, such as coffee, tea, and cookies, will be served. We hope that you can join us.