# Challenge 24 Competition

The Challenge 24 Competition is a mathematics competition for 3rd-, 4th-, 5th-, 6th-, and 7th-graders, hosted each spring by the SBU Department of Mathematics. Participants play the 24 Game, which involves a deck of cards, each showing four whole numbers between 1 and 9. When presented with a card, the players strive to be the first to combine the four numbers to make 24 using only addition, subtraction, multiplication, or division. Many cards have more than one solution. The game can be surprisingly challenging, but the participants in Challenge 24 are quite good, having previously won competitions within their own schools. The coordinator for the SBU Challenge 24 Competition is Dr. Doug Cashing.

Find three solutions... |

The 24 Game was developed in 1988 by Robert Sun to provide students with a fun way to discover patterns among numbers. Sun founded the company Suntex International Inc. to market the game.

Each card in the 24 Game is marked with one dot, two dots, or three dots to indicate its level of difficulty. Generally, the higher the number of dots, the more challenging the card. To the right is a two-dot card containing the numbers 2, 3, 4, and 4. Try making 24 with the numbers on this card. Remember: you must use each of the numbers 2, 3, 4, and 4 exactly once and the only operations you may use are addition, subtraction, multiplication, and division. (Since you must use each of the numbers 2, 3, 4, and 4 exactly once, you will end up using 4 twice.) You do not need to use all four of the operations and you may use an operation more than once. There are three essentially different solutions for this card. Can you find all of them? Clicking the buttons below the card will reveal these solutions.

### To SBU Students: Help!

The Department of Mathematics needs about 30 volunteers to run the Challenge 24 Competition. If you would like to work with kids who love math, sign up to help us! Elementary education majors are particularly encouraged to volunteer. The competition is held on a Saturday in March or early April. The next competition will be planned in the fall, at which time volunteers will be solicited by Dr. Cashing by email and in mathematics classes.

### The 2014 Challenge 24 Competition

The 2014 Challenge 24 Competition was held on Saturday, April 5. There were forty-five students from seven schools attending this year. The participating schools were Austin Area, Cuba-Rushford Middle School, East View Elementary (in Olean), Hinsdale, Olean Intermediate Middle School, Otto-Eldred Elementary School, and Portville. Medals, in the traditional colors of gold, silver, and bronze, were awarded to the first, second, and third place students in each grade. Congratulations to all!

3rd Grade | |

First Place | Dylan DiRosa (East View Elementary) |

Second Place | Emma Dwaileebe (East View Elementary) |

Third Place | Will Snyder (East View Elementary) |

4th Grade | |

First Place | Maryam Mirza (Olean Intermediate-Middle School) |

Second Place | Madison Jones (Olean Intermediate-Middle School) |

Third Place | Hannah Nelson (Olean Intermediate-Middle School) |

5th Grade | |

First Place | Jason Copella (Olean Intermediate-Middle School) |

Second Place | Yuki Wada (Olean Intermediate-Middle School) |

Third Place | Shaini Lal (Olean Intermediate-Middle School) |

6th Grade | |

First Place | Averi Saulter (Cuba-Rushford Middle School) |

Second Place | Sophie Sova (Olean Intermediate-Middle School) |

Third Place | Mackenzie Windus (Olean Intermediate-Middle School) |

7th Grade | |

First Place | Hira Mirza (Olean Intermediate-Middle School) |

Second Place | Lizzie Miller (Portville) |

Third Place | Charity Roulo (Portville) |

Previous Challenge 24 Competitions (participating schools & winners)

The student organizers for the 2013 competition were Margaret Atherton, Amrita Maniram, Esther St. Cloud, Rebecca Tallman, and Steven Zimmer. These students were invaluable assistants to Dr. Cashing in the planning and execution of Challenge 24. Thank you!

### Behind (and within) the Scenes at Challenge 24

On a cold Saturday morning in March, you might expect elementary school and middle school students to be watching TV, texting their friends, or playing outside among the remnants of winter. You may be mostly right---but not completely. In western New York, some 3rd-, 4th-, 5th-, 6th-, and 7th-graders have traveled with parents and teachers to St. Bonaventure University to compete in a math competition. That's a rather beautiful thing.

The Challenge 24 Competition is organized by Dr. Doug Cashing and about five Bona's students. The organizers are aided by about 30 volunteers, who range from SBU students to SBU faculty to middle school teachers to parents. The organizers and volunteers are here for one reason: to celebrate the wonderfully-talented elementary and middle school students who love math so much that they spend a Saturday morning in a math competition. Challenge 24 also celebrates the teachers and the parents who coached and supported their students and children. The following photos of Challenge 24 were taken over the years and together chronicle a single competition.

### Challenge 24: Prelude

**
Prior to the first round,** a student organizer gives the proctors last-minute

instructions in the Doyle Dining Room (above), while coordinator Dr. Doug Cashing

answers questions from students and parents in the adjacent Trustees Room (below).

### Challenge 24: Individual Round

**
During the Individual Round** (seen in the photos above below), students

compete against the deck and the clock. As this round progresses, the room is

is suffused with the murmur of arithmetic operations. The proctors are amazed

by the mathematical skills demonstrated by the participants in Challenge 24.

The Individual Round requires a large number of proctors; some parents and

teachers lend a hand to help the round proceed more efficiently. Thank you!

### Challenge 24: Between Rounds

**
Proctors stand at the ready** seconds before
students enter the room for the

next round of competition.

### Challenge 24: Group Round

**During the Group Round**, students within each grade level compete against

one another in groups of four. In the photo above, a student gives his solution as

his fellow competitors and the two proctors listen intently.

### Challenge 24: Group Finals

**
Challenge 24 culminates with the Group Finals**, in which the top six students

from each grade level compete against one another. Rather than solve a single card,

students must now supply a missing number to solve two cards simultaneously.

The photo above shows the Group Final for grade 4.

### Challenge 24: The Award Ceremony

**
Medals, in the traditional colors of gold, silver, and bronze**, are presented to the

first, second, and third place students in each grade level. Subsequently, photos of the

winners are printed in the Olean Times Herald.

### Challenge 24: Display

**
Around the time of the Challenge 24 Competition**, Dr. Hill puts up a display on

a large bulletin board near the Mathematics Suite in De La Roche Hall. The display

describes the 24 Game and the Challenge 24 Competition and includes fun

facts about the number 24 (see below). It features enlargements of several 24 Game

cards, arranged in the form of a giant "24", so that curious passersby can play the game.

### Fantastic Twenty-Four: Facts about the 24 Game's Magic Number

The most challenging Challenge 24 There are many sets of four numbers between 1 and 9 that cannot be "solved" in accordance with the rules of the 24 Game. For example, the set {1,1,1,1} isn't solvable, because the numbers are too small to reach 24 using the four operations of arithmetic. Of course, 24 Game cards show only solvable sets of four numbers. However, there are some solvable sets of numbers that do not appear on any 24 Game card because they're just too difficult. The two most challenging solvable sets of four numbers for the 24 Game are | |

Divide and conquer 24 has eight different divisors: 1, 2, 3, 4, 6, 8, 12, and 24. That's a lot of divisors for a small number. This may be why Robert Sun chose 24 as the "target" number for his game. The more divisors a number has, the easier it is to make the number from a product of two numbers. There are four products that equal 24: 1x24, 2x12, 3x8, and 4x6. The lowest integer possessing more divisors than 24 is 36. | |

Just the factorials, ma'am 24 is 4 factorial, written 4!. The factorial of a positive integer n is the product of the integers from 1 up to n. Thus, 4! = 1x2x3x4 = 24. Factorials are fantastically useful in the theory of counting. For example, suppose the editor of the Olean Times Herald is arranging the four photos (one for each grade) of the Challenge 24 winners. She plans to place them in a single row. She could order them in the usual way (4th grade, 5th grade, 6th grade, 7th grade) or she could be creative and order them some other way. The number of ways the editor can order the four photos in a row is 4! = 24. | |

More fun factorials 24 factorial, written 24!, equals 620448401733239439360000, which has precisely 24 digits. The only other positive whole numbers n for which n! has exactly n digits are 1, 22, and 23. | |

One day at a time There are 24 hours in a day---a fact that allowed "24" to be the title of a popular TV series. The 24-hour day seems to have originated with the ancient Egyptians. The Egyptians liked counting in base 12, perhaps because the human hand has 12 finger joints. (A hand has four fingers and a thumb; each finger has three joints.) With one hand, you can use your thumb to count to 12 on your finger joints. (Try it and see.) Base-12 seems rather odd to us, but is it really any stranger to use a base-12 number system because a hand has 12 finger joints than it is to use a base-10 system because two hands have 10 fingers (well, eight fingers and two thumbs)? Given their affection for base-12, it was natural for the Egyptians to divide the day into 12 parts and the night into 12 parts, which led to the modern 24-hour day. | |

It's worth a shot The NBA uses a 24-second shot clock to time possessions by the offensive team. The shot clock was invented by Danny Biasone, the late owner of the Syracuse Nationals, in the 1950s to try to speed up the game and prevent teams from stalling. Why 24 seconds? Biasone figured that the average number of shots two teams would take during a game was 120. He divided that number into the length of a game, which is 48 minutes or 2,880 seconds. The result is 2880/120 = 24 seconds. | |

Far side of the Moon As of 2009, the 40th anniversary of the first manned landing on the Moon, exactly 24 human beings have travelled to the Moon. "Travelled to" means "reached Moon orbit or landed on the Moon". Of these 24, 12 walked on the Moon. Thus far, all manned missions to the Moon occurred between 1968 and 1972. | |

License to derive You can play the 24 Game using New York license plates! The license plates for most NY cars have the format LLL-DDDD, where L is a letter and D is a digit between 0 and 9. The four digits can, in principle, be used for the 24 Game. However, unlike the 24 Game cards, some license plates have one or more zeroes and it's not always possible to make 24 from the digits on a license plate. Nevertheless, the game can be successfully played with the license plates of many---perhaps most---NY cars. Portions of a few NY plates appear below. Speaking of cars.. | |

A good sign The Interstate 86 exit closest to St. Bonaventure University is Exit 24, as seen on the sign below. |