𝐃𝐚𝐲 𝟒 𝐚𝐭 𝐑𝐀𝐌 𝐓𝐏 𝟐𝟎𝟐𝟓: 𝐀𝐝𝐯𝐚𝐧𝐜𝐢𝐧𝐠 𝐌𝐚𝐭𝐡𝐞𝐦𝐚𝐭𝐢𝐜𝐚𝐥 𝐅𝐫𝐨𝐧𝐭𝐢𝐞𝐫𝐬 𝐍𝐮𝐦𝐛𝐞𝐫 𝐓𝐡𝐞𝐨𝐫𝐲 𝐃𝐫. 𝐌𝐮𝐫𝐚𝐥𝐢𝐝𝐡𝐚𝐫𝐚𝐧 led an in-depth session on number theory, guiding students through Thue’s Lemma and Fermat’s theorem on sums of two squares, including Euler’s elegant proof by infinite descent. Participants also examined the impossibility of solutions to x⁴+y⁴=z⁴ and explored Lagrange’s four-square theorem. 𝐆𝐞𝐨𝐦𝐞𝐭𝐫𝐲 𝐚𝐧𝐝 𝐏𝐫𝐨𝐛𝐚𝐛𝐢𝐥𝐢𝐭𝐲 𝐃𝐫. 𝐀𝐬𝐡𝐰𝐢𝐧 𝐆𝐮𝐡𝐚 brought geometry to life with an exploration of conic sections and applied curve fitting to resolve the "squarest country" debate. He also introduced students to probability paradoxes and the application of Bayes’ theorem, challenging conventional intuition. 𝐀𝐥𝐠𝐞𝐛𝐫𝐚𝐢𝐜 𝐒𝐭𝐫𝐮𝐜𝐭𝐮𝐫𝐞𝐬 𝐚𝐧𝐝 𝐀𝐩𝐩𝐫𝐨𝐱𝐢𝐦𝐚𝐭𝐢𝐨𝐧𝐬 𝐁𝐡𝐚𝐬 𝐁𝐡𝐚𝐦𝐫𝐞 continued the exploration of Gaussian integers, highlighting their unique properties. He also demonstrated the practical use of continued fractions for increasingly accurate approximations of √5. 𝐌𝐚𝐭𝐡𝐞𝐦𝐚𝐭𝐢𝐜𝐚𝐥 𝐏𝐮𝐳𝐳𝐥𝐞𝐬 𝐚𝐧𝐝 𝐂𝐥𝐚𝐬𝐬𝐢𝐜𝐚𝐥 𝐏𝐫𝐨𝐛𝐥𝐞𝐦𝐬 𝐇𝐚𝐫𝐚𝐠𝐨𝐩𝐚𝐥 engaged students with innovative number puzzles, a study of partitions, and Diophantine equations. He also discussed the mathematics underlying classic "impossible" problems such as squaring the circle and angle trisection. 𝐂𝐫𝐲𝐩𝐭𝐨𝐠𝐫𝐚𝐩𝐡𝐲 𝐚𝐧𝐝 𝐆𝐚𝐦𝐞 𝐓𝐡𝐞𝐨𝐫𝐲 𝐀𝐚𝐝𝐢𝐭𝐲𝐚𝐧 𝐆𝐚𝐧𝐞𝐬𝐡 examined the intersection of cryptography and game theory, focusing on optimal strategies for secure password storage. 𝐂𝐨𝐦𝐦𝐮𝐧𝐢𝐭𝐲 𝐚𝐧𝐝 𝐈𝐧𝐬𝐩𝐢𝐫𝐚𝐭𝐢𝐨𝐧 Each day concluded with an informal discussion hour, fostering connections beyond mathematics. Day 4 was particularly memorable, as teaching assistants shared personal stories of perseverance and ambition, inspiring all participants. Across all sessions, students engaged in rigorous proofs, creative problem-solving, and the excitement of discovering new mathematical insights.