Download 1. Consider the square pyramidal numbers formed by counting

P ROBLEM SET 1
1. Consider the square pyramidal numbers formed by counting objects stacked (3-dimensionally)
in concentric squares. The first few values are 1, 5, 14, 30, . . .. Can you find a formula for these?
Diaz, Nicolas
Miguel
Al Ali, Rahma
Helson, Bethany
Johnson, Derek
Martinez de Andino,
2. Consider the tetrahedral numbers formed by counting objects stacked (3-dimensionally)
in concentric equilateral triangles. The first few values are 1, 4, 10, 20, . . . . Can you find a
formula for these?
Nye, Margaret
Sall, Alison
Spence, Ian
Willemann, Ryan
Dewey, Haley
3. Show that if (j, k) is an integer solution to 1 = j 2 − 2k 2 then so is
(3(j + k) + k, 2(j + k) + k) .
Use this to find a number larger than 36
√ that is both square and triangular, and to find the
corresponding rational approximation to 2.
Hartbarger, David
Hooper, Rachel
Levy, Andrew
Nordike, James
Rivera, Malia
4. Show that 3 is the only prime whose square is the sum of two consecutive cubes (as in,
3 = 23 + 13 ).
2
Smith, Brian
Wasson, Emily
Bakirdan, Volkan
Fuller, Andrew
Hill, Ian
5. Suppose we want to pack cubical boxes (with all sides having the same length) into a
rectangular solid measuring a feet long, b feet wide, and c feet high. If a, b, and c are all
integers then we can always use 1 × 1 boxes but sometimes we can do better; for example, if a,
b and c are all even then we can use 2 × 2 boxes.
What is the largest dimension for the cubic boxes that we can use to completely pack the
space? Describe an algorithm to actually find this number, given a, b, and c.
Jones, Rachel
Lydia
Miklos, Savannah
Rhodes, Ben
Scranton, Kendall Olivia
Washington,