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UMBRELLA ILLUSION

One of my illusions from the late ’90s. Take a look at the colorful umbrellas in Figures A and B—are they the same or different? About 80% of people will say that Umbrella A has jagged, zigzag edges, while Umbrella B has smooth, wavy lines. But here’s the trick—you’ve been fooled by the brightness contrast of the rays inside the umbrellas. In reality, both umbrellas are identical in shape, perfectly congruent.

This illusion shows a phenomenon called curvature blindness, which was rediscovered in 2017 by Japanese psychologist Kohske Takahashi. He created a powerful variant and studied its impact on how we perceive shapes.

© Kohske Takahashi – The wavy lines appear different depending on the background and how the repetitive dark color is applied to them.

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The Zaniest Word in Math: Zenzizenzizenzic

One of the most peculiar numeral words in English, zenzizenzizenzic (/’zɛnziːzɛnziːzɛnzik/), denotes the square of the square of a number’s square. It appeared only once in English, in Robert Recorde’s The Whetstone of Wit (1557). The term derives from the obsolete zenzic, meaning the square of a number. Zenzic was borrowed from German, where mathematicians of the 14th and 15th centuries adopted it from the medieval Italian censo, itself a descendant of Latin census. Italian algebraists used censo to translate the Arabic māl (literally “possessions” or “property”), the standard term for a squared number. This association arose because early mathematicians, including the Arabs, conceptualized squared numbers as representing areas, particularly land—hence, property.

Notably, zenzizenzizenzic is the only English word with six Zs.

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How a Schoolboy Outsmarted a Tedious Task

1780s, Germany. A schoolteacher, desperate for some peace, gives his 8-year-olds a tedious task: add up all the numbers from 1 to 100. That should keep them busy, right?

Enter young Carl Friedrich Gauss. While his classmates grind away, he takes a quick look, “folds” the numbers—1 pairs with 100, 2 with 99, and so on—realizing each pair sums to 101. With 50 such pairs, he multiplies: 50 × 101 = 5050.

Boom. Two minutes, problem solved. Teacher stunned. Classmates still counting. Gauss goes on to become one of history’s greatest mathematicians.

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Creating Perfect Squares from Odd Integers

It’s visually easy to see that the sum of consecutive odd numbers forms perfect squares—this brilliant animation is perfect for empirically understanding why. But how can we explain it in words?

1️⃣ The sum of consecutive odd numbers produces perfect squares
– The sequence of odd numbers:
1, 3, 5, 7, 9, …
– The sum of the first n odd numbers follows the formula:
1 + 3 + 5 + … + (2n-1) = n²
– This can be proven by induction.

2️⃣ The discrete derivative of n² is 2n + 1
– The discrete derivative (forward difference) of a function f(n) is:
Δ f(n) = f(n+1) – f(n)
– Applying it to f(n) = n²:
(n+1)² – n² = n² + 2n + 1 – n² = 2n + 1
– This shows that the difference between consecutive squares is always an odd number—specifically, the (n+1)th odd number!

A simple—well, for those who love math—yet beautiful mathematical insight!

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Geometry Meets Illusion

A geometrical optical illusion to explore with your kids!

a. The illusion is created by context.

b. Here, the key factor is perspective.

c. Conclusion: When two objects are the same size, the one that appears farther away will look larger.