Abstract In this work, we will give a novel method to construct a continuous approximation of the Exponential, Logistic, and Gaussian functions that allow us to do a handmade drawing of their graphs for which there is no accuracy of drawing at elementary levels (even at advanced ones!). This method arises from solving the elementary ordinary differential equation x0 (t) = ax(t) combined with a suitable piecewise constant argument. The proposed approximation will allow us to generate several numerical schemes in an elementary way, generalizing the classical ones as, Euler’s schemes. No sophisticated mathematical tools are needed.

Probability is a rich part of applied mathematics. First we have a look at the unique encounter of two 23 year old chess players on their birthday which leads to an interesting set of questions. Next some chess tournaments lead us to partition numbers, multinomial coefficients and Bell numbers. After that we arrive at the Pentagonal Sea where the beautiful structure of Euler's ship leads to a new look at the recurrence relation of the partition function and the pentagonal number theorem. Aboard Euler's ship we find the Lazy Caterer's and the Connell sequences. At the end we look at the probability that the above mentioned chess confrontation will ever happen again and tell you how to decide whether to accept or reject a wager.