Four double-multiplication problems that intuitively trick your thinking.

The world around us is full of wonders, and sometimes intuition deceives our thinking. Below are four seemingly easy math problems that could be solved in a few minutes, but are actually quite complex.

1. The problem of arranging grains of rice on a chessboard.

In the 6th century, the King of India offered Seta, the inventor of chess, gold and silver as a reward, but Seta refused and wanted to be rewarded with grains of rice in the following way: "Place one grain of rice in the first square, two in the second, four in the third, and so on, doubling the number of grains in each subsequent square until the entire 64-square chessboard is filled."

The King accepted, but didn't forget to sarcastically remark that Seta had missed an opportunity to get rich.

However, the next day, the King realized his mistake because the number of grains of rice was terrifyingly large: 1 + 2 + 2² + ... 2⁶² + 2⁶³ = 2⁶⁴ - 1 = 18,446,744,073,709,551,615

This amount of grain was millions of times greater than the King's current supply and could cover the entire surface of the earth. Knowing that there wouldn't be enough grain to reward him, but to keep his promise, the King heeded the wise man's advice and ordered: "Seta, you yourself must count each grain of grain precisely."

According to calculations, it would take 60,000,000,000 years to count all the rice grains, and if each granary were 4 meters high and 10 meters wide, the total length of all these granaries, when lined up end-to-end, would reach 300,000,000 kilometers—twice the distance from Earth to the Sun.

2. The paper folding problem and the 2002 Guinness World Record

Try folding a thin A4 sheet of paper in half repeatedly, and you'll see you can only fold it a maximum of 7 times! Because after the 8th fold, you'll have to fold a 256-page book in half.

To achieve even greater feats, in 2002, Britney Gallivan, a high school student in the US, chose a 0.1 mm thick, 1,219 m long piece of tissue paper and spent eight hours crawling down a long aisle in a California shopping mall to fold it in half twelve times consecutively. She was subsequently recognized by the Guinness World Records for the most times a single piece of paper was folded.

Britney Gallivan folded a 1,219-meter-long strip of paper into 4,096 layers, setting a Guinness World Record. Photo: Guinnessworldrecords

As we continue with the calculations, we will see the tremendous power of exponentiation, even with a base of 2 - the smallest natural number greater than 1.

With a paper thickness of 0.1 mm, after the nth fold, the thickness will be 2 to the power of n x 0.1 mm. More specifically, after the 12th fold, the paper is as thick as a chair, but after the 17th fold, it is as thick as a two-story building.

After 42 folds, the paper would be 439,800 km thick – greater than the distance from Earth to the Moon (384,400 km). Each time it's doubled, the thickness doubles while its surface area is halved. After 51 folds, the paper strip would be longer than the 200 million km distance from Earth to the Sun. And after 103 folds, the ultrafine paper strip would be over 100 billion light-years long, larger than the diameter of the observable region of space, which covers approximately 93 billion light-years (at the speed of light, 300,000 km/second).

3. The dilemma of sons-in-law choosing dowry in 2017

In 2017, India hosted the 19th International Junior Mathematical Olympiad (InIMC). Knowing that Indian marriage ceremonies are very different from those in other countries, I created a fun math problem for the Vietnamese 6th-grade team during their training for the 2007 InIMC.

This problem retains the original idea of ​​doubling the value but is creatively adapted to suit the traditional Indian marriage custom of "the son-in-law receiving a dowry from the bride's family".

4. The problem of the number of people infected with the SARS-CoV-2 virus.

In March 2020, during the Covid-19 pandemic, I set a poem by Dr. Nguyen Manh Thang to music, creating the song "The World Joins Forces to Fight the Coronavirus Pandemic," and a mathematical problem about the growth rate of the SARS-CoV-2 virus in the human body.

The problem is as follows: A person has just been infected with the SARS-CoV-2 virus, and every 3 minutes each virus duplicates into 2 new viruses. Assuming that after 81 minutes of infection, the person's body has 402,653,184 viruses and begins to show symptoms, how many SARS-CoV-2 viruses were initially infected?

Solution guide: This problem has the reverse structure of the previous three problems. To solve it, we will analyze 81 ÷ 3 = 27 and 402,653,184 = 3 × 2 to the power of 27.

Therefore, the answer is that the human body was initially infected with 3 SARS-CoV-2 viruses.

Tran Phuong (Deputy Director of the Talent Development Center)

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