Today is July 14, 2021, which can be written 7-14-21…. so I’m calling it “Sevens Day”! Here are some wonderful things about the number SEVEN and related numbers:
- 7 is the number of days in a week, 14 is the number of days in a fortnight, and 28 is the number of days in a lunar month.
- 7 is the start of an arithmetic progression of six prime numbers: 7, 37, 67, 97, 127, 157. Sadly 187 breaks the sequence since it is a multiple of 11 and 17.
- There is only one way that 7 can be decomposed into 3 unique addends, and they are consecutive multiples of 2: 1 + 2 + 4 = 7. This is a useful fact for KenKen and Kakuro puzzles! Other decompositions of 3 addends all include repeated numbers: 3 + 1 + 3 or 2 + 3 + 2 or 1 + 5 + 1.
Divisibility* by 7 is something I learned when doing more advanced KenKen puzzles. The technique I use is this:
- Chop off the units’ digit and double it.
- Subtract this from the remaining number.
- Continue this process until the result is 1 or 2 digits.
- If the final number is divisible by 7, then the original number was divisible by 7 [Note that 0 is divisible by 7].
For example: 3052
- Take the 2 and double it to get 4.
- 305 – 4 = 301
- Take the 1 and double it to get 2.
- 30 – 2 = 28 which is divisible by 7.
Another version of a divisibility test for 7s was discovered** by Chika Ofili, a 12-year-old Nigerian boy, in 2019. His version is:
- Chop off units’ digit and multiply by 5
- Add this to the remaining number
- Continue… if the result is divisible by 7, then the original was divisible by 7.
This has come to be called “Chika’s Test”. It might be faster than my other method, because it is faster to get to a recognizable multiple of 7 ( a number between 0 and 70) by adding multiples of 5 than subtracting multiples of 2. And adding numbers is sometimes simpler than subtracting numbers. Read more about Chika’s test here.
Why do these methods both work? For example, take the number with digits ABCD. Chop off units’ digit D, this leaves ABC which has value 100A + 10B + C.
Adding 5D vs. subtracting 2D is only changing the result by a multiple of 7, so if the number was divisible by 7, it still is. If it wasn’t, it still isn’t.
[100A + 10B + C + 5D] – [100A + 10B + C – 2D] = 7D.
Seven is also a special number for mathematical Symmetry: there are exactly 7 Frieze Patterns. This page gives a nice visual with some simple explanations: Frieze Patterns.
To dive more deeply, check out Paula Beardell Krieg’s (@PaulaKrieg) series of 10 posts, including lots of artwork and models. Start here for an introduction, and this link gives access to all of the posts.
Fraction Patterns with Sevenths: Any math teacher will tell you that fractions with 7 in the denominator do not have simple decimal conversions. The repeating decimals aren’t as “friendly” for students as those that happen with thirds, sixths, or ninths.
But the repeating pattern is quite beautiful, with 6 different digits in the pattern:
1/7 = 0.142857… = 0.142857142857142857142857
What’s even cooler to see is that all the fractional sevenths have the same 6 digits in the same order, each one starting with the next higher value (of 1, 2, 4, 5, 7, 8):
2/7 = 0.285714…
3/7 = 0.428571…
4/7 = 0.571428…
5/7 = 0.714285…
6/7 = 0.857142…
Read more about these “Magic Sevenths” here.
Heading over to Geometry, the regular polygon with 7 sides (known as a heptagon or septagon) is the smallest polygon that can’t be constructed with a ruler & compass alone.
This week, Becky Warren (@becky_k_warren) shared the method of “tricking” GeoGebra into rotating a segment exactly 360/7° to create a heptagon.
One more fun fact I uncovered this morning has to do with right triangles and Pythagorean triples. If a and b are the two shorter sides of a Pythagorean Triangle (a right triangle with all integer sides, that is, a set of lengths that comprise a Pythagorean Triple), then one of these will be divisible by 7: a or b or a+b or a-b. Check it out with your favorite Pythagorean Triples!
Have a wonderful Sevens Day!!
* More on Divisibility and math-y dates in my post Leap Years & License Plates; there are some nice references on divisibility rules at the bottom.
**Chika’s divisibility test wasn’t a new discovery (it is listed in the Penguin Dictionary referenced below, for example), but it was new to him and quite a feat for a 12-year-old!
Some of these facts are from The Penguin Dictionary of Curious and Interesting Numbers by David Wells (1986).
While searching for images for this post, I learned about electronic 7-segment displays, used in many applications to display numbers (and some letters). Did you have a digital clock that looked like this?