{'fraction_non_alphanumeric': 0.015288394718554551, 'fraction_numerical': 0.010423905489923557, 'mean_word_length': 4.603112840466926, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Young Students Approach Integer Triangles

How many different triangles of perimeter n can be made with a supply of n toothpicks? Let us assume all triangles created have an integral number of toothpicks per side and have positive area. We wish to count the number of incongruent triangles that can be so constructed. This is the problem students aged 12 17 grappled with over ten one-hour sessions of The Math Circle in the fall of 2001. Organized as a school for the enjoyment of mathematics, The Math Circle offers programs of courses for young students, aged 5 through 18, who enjoy math and want more than the typical school curriculum offers. We hold classes at Harvard University and Northeastern University, and at community education centers. The school was founded by Robert Kaplan, Ellen Kaplan and Tomas Guillermo in 1994, and I have been fortunate to be involved with this program for the past two years. We typically begin a semester by posing a mathematical problem to each of our classes and allowing the small groups of students to explore the mystery for the ten weeks that follow. The role of an instructor in The Math Circle is not so much to instruct as to guide and offer occasional hand-holds (or sometimes to throw spanners into the works). We want our students to experience for themselves the creative aspects of mathematics, to make mistakes, to experience the frustrations of doing math, as well as the deep joy