Category: Numbers

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Visual Math Challenge: Rectangle in an Octagon

One fascinating property of a rectangle inscribed in an octagon is that its side ratio aligns perfectly with the “silver ratio“, or 1+√2. But there’s more to discover! Without doing any calculations, can you prove that the area of this gray rectangle is exactly half of the full octagon’s area? Give it a try!

The “silver ratio” is connected to various mathematical concepts, such as Pell numbers and continued fractions. It serves as the limiting ratio of consecutive Pell numbers, similar to how the golden ratio relates to Fibonacci numbers.


show solution

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Discover the Golden Ellipse

A golden ellipse is one where the axes are in golden proportion, meaning the ratio of the major axis (a) to the minor axis (b) is the golden ratio:
φ = (1 + √5)/2.
To visualize this, draw a golden ellipse along with its inscribed and circumscribed circles: the smallest circle fitting inside the ellipse and the largest circle surrounding it.
Interestingly, the area of the ellipse matches the area of the “annulus” formed between these two circles!
Here’s how it works:
Let a be the semi-major axis and b the semi-minor axis, with a = φb.
The area of the annulus is:
π(a² − b²) = πb²(φ² − 1)
The area of the ellipse is:
πab = πφb²
And as φ² − 1 = φ, then πb²(φ² − 1) = πφb².


Isn’t it fascinating how geometry intertwines with the golden ratio?

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Ambigram Magic Squares

When using standard characters, the digits 0, 1, and 8 are symmetrical around the horizontal axis, while 6 and 9 are interchangeable when rotated 180 degrees.

With these digits, we can create magic squares that maintain their constant sum even when flipped, as illustrated below.

Interestingly, when these numbers are represented in LCD style, we can also include the digit 2, which resembles a 5 when inverted. This allows for the creation of magic squares with additional properties related to both 2D and 3D symmetry—whether flipped or mirrored—such as the remarkable example created by Chris Wardle.

This isn’t the first magic square to exhibit such fascinating properties; there are many variations out there. I encourage you to explore and discover your own creations online. If you have original ideas for magic squares with these unique characteristics, we would love for you to share them with us!

For those interested in learning more about the history and mathematics behind magic squares, check out this fascinating article from the Royal Institution: The Fascination of Magic Squares.

More links of interest:
[1] https://www.rigb.org/explore-science/explore/blog/fascination-magic-squares
[2] https://math.hmc.edu/funfacts/magic-squares-indeed/
[3] https://patcherymenagerie.blogspot.com/2019/07/magic-squares.html
[4] https://www.geeksforgeeks.org/magic-squares-fun-fact-and-more/
[5] https://mathcommunities.org/magic-squares/
[6] https://www.magischvierkant.com/links-eng/
[7] https://chelekmaths.com/2020/06/30/cracking-the-cryptic-joy-and-magic-squares/
[8] https://www.byrdseed.tv/magic-squares/

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Numeronyms in Chinese

Numeronyms, a fascinating aspect of Chinese internet slang, utilize numbers to represent phrases or concepts based on their phonetic similarities. This shorthand is commonly used in online communication, allowing users to convey messages succinctly and creatively. Below are some popular Chinese numeronyms and their meanings:

  • 007 – Represents a work schedule of 00:00 hours to 00:00 hours, 7 days a week (pinyin: línglíngqī), a variant of the 996 working hour system.
  • 1314 – Means “forever,” often used in romantic contexts, as in “I love you forever.” It corresponds to 一生一世 (pinyin: yīshēng yīshì).
  • 213 – Refers to “2B,” which is a derogatory term for someone perceived as very stupid (二逼).
  • 233 – Represents laughter, equivalent to 哈哈哈 (pinyin: hāhāhā).
  • 250 – Means “stupid,” derived from historical currency references (pinyin: èrbǎiwǔ).
  • 38 – Refers to a woman who gossips excessively.
  • 4242 – Indicates “yes” or “affirmative,” representing 是啊是啊 (pinyin: shìa shìa).
  • 484 – Stands for “if,” corresponding to 是不是 (pinyin: shìbùshì).
  • 520 – Means “I love you,” as it sounds similar to 我爱你 (pinyin: wǒ ài nǐ).
  • 555 – Represents crying, mimicking the sound 呜呜呜 (pinyin: wūwūwū).
  • 666 – Indicates something “cool” or “nice,” derived from gaming slang where it signifies impressive skill (pinyin: liùliùliù).
  • 777 – A variation of 666, meaning “even better.”
  • 7451/7456 – Expresses anger, translating to 气死我了 (pinyin: qìsǐwǒle), meaning “I’m furious.”
  • 748 – An aggressive phrase meaning “Go and die!” (去死吧, pinyin: qùsǐba).
  • 87 – Refers to being “bitchy” or “idiotic” (白痴, pinyin: báichī).
  • 88 – A casual way to say “bye bye” (拜拜, pinyin: bābā).
  • 94 – Functions as a conjunction, meaning “so” or “but” (就是, pinyin: jiùshì).
  • 955 – Represents a typical 9 to 5 job (pinyin: jiǔwǔwǔ).
  • 99 – Symbolizes the wish for a couple to be together for a long time (久久, pinyin: jiǔjiǔ).
  • 995 – A plea for help, meaning “Save me!” (救救我, pinyin: jiùjiùwǒ).
  • 996 – Refers to the demanding 996 working hour system (pinyin: jiǔjiǔliù).
  • 999 – Similar in meaning to “666,” often used to express approval.

These numeronyms reflect the creativity and efficiency of language use in the digital age, showcasing how numbers can convey complex emotions and ideas in a concise manner.

⇨ Further reading.

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When Intuition Fails: Borwein Integrals

Borwein integrals, introduced by David and Jonathan Borwein in 2001, are a classic example of how intuition can be misleading in mathematics. Defined using the sinc function (sin(x)/x), these integrals initially appear to consistently equal π/2. However, this pattern unexpectedly breaks down when the factor sinc(x/15) is introduced, surprising even experienced mathematicians. This phenomenon underscores the importance of rigorous proof over reliance on patterns.


Read further. 


[David Borwein and Jonathan M. Borwein, “Some Remarkable Properties of Sinc and Related Integrals,” Ramanujan Journal 5:1 (March 2001)]

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Magic Square for Gunners

A magic square is a grid where the sum of the numbers in each row, column, and diagonal is the same, creating a harmonious balance. A “geomagic” square, on the other hand, is a grid of geometrical shapes where each row, column, or diagonal can be assembled into an identical shape known as the “target shape”. Like numerical magic squares, all shapes in a geomagic square must be distinct.

Concept by Lee Sallows.

The postage stamp below, issued by Macau Post on October 9, 2014, pays tribute to Lee Sallows, the creator of geomagic squares.

Further reading.

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Why does the symbol for zero look like a capital ‘o’ (0)?

The history of this fundamental number is surprisingly intricate. Its roots can be traced back to ancient Babylon around 300 BCE, where a positional numeral system employed two slanted wedges to signify an empty place in a number. However, this was merely a placeholder without any numerical value.

A more concrete step towards our modern zero emerged in Greece. The letter omicron (ο), short for οὐδὲν (ouden = nothing), was utilized as a placeholder in astronomical calculations by figures such as Ptolemy and Iamblichus as early as the 1st century CE. This practice likely influenced Indian mathematicians following Alexander the Great’s conquests.

Indian mathematicians revolutionized this concept by transforming the placeholder into a full-fledged number. Initially represented by a dot called ‘bindu‘ (बिन्दु), zero became a cornerstone of arithmetic and algebra. This innovation was crucial in the development of our modern number system.