MATH in the News

Each week we'll post a math problem based on a current news story. Read the story and solve the problem.

Does this expression look happy or sad?
Detail: Leonardo da Vinci (1452-1519), Mona Lisa (La Gioconda), 1503-1506. Oil on wood, 30.3x20.9in. (77x53cm). Musée du Louvre, Paris. ©RMN-Grand Palais/Art Resource, NY. Photo: Michel Urtado.
Mona Lisa’s Smile
By Paul Cates
Combinations: Researchers use a scientific approach to get to the bottom of the world’s most famous grin

For centuries, people have debated about the Mona Lisa, a 16th-century painting by Leonardo da Vinci. They wondered how to interpret the expression on the woman’s face. Some viewers say she is smiling but others think she looks sad. Some people even feel that the woman’s look, if viewed for a long time, changes from one of joy to one of sadness. Now science may have finally solved the riddle of Mona Lisa’s true emotion.

Neuroscientists, scientists that study the brain, at the University of Freiburg in Germany decided to test how viewers interpret the woman’s expression. Because the Mona Lisa, is arguably the most recognizable painting in the world, the scientists had to develop a creative way to conduct their experiment. People who had already seen the image might be biased and already think about the painting in a particular way.

First, the researchers created twelve digital copies of the painting. Each had a small change to the woman’s smile. In each version, the scientists altered the famous mouth to turn it up or down at the corners in varying degrees. That made the Mona Lisa’s, expression seem happier or sadder.

Next, the scientists showed 9 of the images, chosen from all 12 altered copies and a copy of the original painting, to a group of people. The people recorded what emotion they saw in each image. To make sure a person’s responses stayed the same, the scientists showed the images to each viewer 30 times in random order.

According to the study’s results, people perceive Leonardo’s Mona Lisa, to be smiling an overwhelming 97 percent of the time. The study revealed that people see the woman as joyful. It also shows how the human brain interprets emotion. These findings will have a long-term effect on the art world and the scientific community.


Of the 12 variations of the Mona Lisa, 6 of her smiles were altered to look sad and 6 had happier smiles than the normal Mona Lisa. Only 9 options were shown to each participant at a time. If 4 of the 9 images shown to one participant were sad expressions, how many unique combinations of the sad expressions were possible?

Check Your Answer

You have 6 choices to make a set of 4 sad expressions: ML1, ML2, ML3, ML4, ML5, ML6. Here are the different unique combinations you can make:

ML1 – ML2 – ML3 – ML4
ML1 – ML2 – ML3 – ML5
ML1 – ML2 – ML3 – ML6
ML1 – ML2 – ML4 – ML5
ML1 – ML2 – ML4 – ML6
ML1 – ML2 – ML5 – ML6
ML1 – ML3 – ML4 – ML5
ML1 – ML3 – ML4 – ML6
ML1 – ML3 – ML5 – ML6
ML1 – ML4 – ML5 – ML6
ML2 – ML3 – ML4 – ML5
ML2 – ML3 – ML4 – ML6
ML2 – ML3 – ML5 – ML6
ML2 – ML4 – ML5 – ML6
ML3 – ML4 – ML5 – ML6

So there are a total of 15 possible unique combinations.