PinnedMember-onlyAbstract Algebra: Seeing the Same Thing in Different PlacesStructures behaving similarly — What types of things are studied in Abstract Algebra? Below we will discuss three seemingly different things which in the end all behave in the same way. It is common to see this as results from one branch of Mathematics are often used to say things about other branches. Structure 1 — Maps on a set of three elements Imagine…Mathematics5 min read

Apr 19Member-onlyDemonstrate There Is An Irrational Number Between Any Two Distinct Real NumbersIrrational numbers are ones that cannot be written as an integer fraction p/q where p and q≠0 are integers — Most of us have seen countless resources explaining how numbers such as √2, √3, e=2.71828…, and others are irrational. To recap, a number is said to be rational if it can be written as an integer fraction p/q where p and q≠0 are integers e.g. 1.23=123/100 is a rational number…Education3 min read

Apr 18Member-onlyArgue In Your Head m³-m Is Divisible By 6 For Any Integer mBrain Exercise — Notice the following are all multiples of 6: 2³-2=6=6 times 1 3³-3=24=6 times 4 4³-4=60=6 times 10 5³-5=120=6 times 20 6³-6=210=6 times 35 7³-7=336=6 times 56 (-2)³-(-2)=-6=6 times -1 How could we argue m³-m is divisible by 6 for ALL integers m? One way is to observe m³-m=m(m²-1)=m(m-1)(m+1). This must…Education2 min read

Apr 17Member-onlyChallenge: Write Down All 4-Digit Palindromic Numbers Divisible By 42Palindromic Numbers Read The Same When Reversed — A palindromic number is one which reads the same when reversed e.g. 11, 12321 , 12455421. In this piece we restrict our attention to 4-digit palindromic numbers and ask which of these are divisible by 42? First, we note if a number is divisible by 42 then it is divisible…Education3 min read

Apr 7Member-onlyProve √6-√2 > 1 Without Estimating √6 or √2Exercise In Logic — We can obviously use a calculator (or Google) to check that √6-√2 > 1 since √6=2.4494…. and √2=1.4142…, but how could we prove such an inequality without checking these values? One tool available to mathematicians is ‘a proof by contradiction’. Here we assume the opposite statement (in this case √6-√2…Education2 min read

Apr 6Member-onlyHow Many Positive Numbers Below 7 Million Contain The Digit 7 And Are Divisible By 7?Exercise In Logic — The numbers 7, 70, 77, 147, 175 and 217 are the first six positive numbers which contain the digit 7 and are divisible by 7. How many are there below 7 million? For clarity, let’s not include the number 7 million. Let’s first ask a more basic question: how many…Education3 min read

Published in Math Simplified·Apr 4Member-onlyWhy Do The Vertices Minus The Edges Plus The Faces Always Equal 1 (V-E+F=1) For Graphs Like This?On Connected Plane Graphs — Have a look the diagrams above. We call them plane graphs, which we define as a figure in two dimensions consisting of points (vertices) and edges joining these vertices such that no two edges cross each other. Specifically, the above are examples of connected plane graphs because we can get…Education3 min read

Apr 2Member-onlyRiddle: What Colour Is The Mathematicians Hat?Can You Figure This Out? — Three mathematicians stand in a line, one in front of each other. The first cannot see the other two, the second can see the first but not the third, and the third can see the other two. The mathematicians are shown five hats, two brown and three white. One hat…Education2 min read

Published in Math Simplified·Apr 1Member-onlyClassic Puzzle: Find The Counterfeit Coin Amongst 12 Identical Looking CoinsOnly For The Brave — Suppose you have twelve identical-looking coins, where one coin is counterfeit and is either heavier or lighter than the others. You are given a balance scale to find the counterfeit coin. You can only use the scale three times, how will you find the counterfeit? …Education4 min read

Published in Math Simplified·Mar 31Member-onlyWrite Down All 3-Digit Palindromic Numbers Divisible By 18Skip work this morning — A palindromic number is one which reads the same when reversed e.g. 11, 12321 , 12455421. In this piece we restrict our attention to 3-digit palindromic numbers and ask which of these are divisible by 18? First, we note if a number is divisible by 18 then it is divisible…Education2 min read