The Four Fours - an Ancient Problem

Date: 10/02/97 at 07:32:00
From: Alec
Subject: The Four Fours - ancient problem

My class is trying to solve the problem of the Four Fours. The problem 
allows you to use only four fours and as many operands as you like in 
order to create mathematical sums which equal the numbers from 1 to as 
high as you can go. While it is possible to create some quite high 
numbers, you must have all the previous numbers before these become 
part of the answer. Can you help me by seeing how many you can get or 
should I only ask for the ones we cannot find?

Alec Tibbitts

Date: 10/09/97 at 23:49:39
From: Doctor Chita
Subject: Re: The Four Fours - ancient problem

Hi Alex:

I'm quite familiar with this problem. I was a math teacher and once 
when my husband and three children and I were driving to Florida 
(a 24 hour trip), playing this game took us through at least two 
states. I think we were stopped at 37 or thereabouts.

I found that you could generate more of the more "difficult" numbers 
by knowing a some specialized math functions. To my family's dismay 
and annoyance, I used the greatest integer function. That is, the 
greatest integer value of an integer is the integer itself. The 
greatest integer value of a decimal number is the greatest integer 
immediately to the left of the decimal number. For example:

  a. [3] = 3

  b. [2.6] = 2

  c. [-5.4] = -6

  d. [-4] = -4

This function obviously helps to truncate a number.

You could also introduce the rounding function, using a rule about 
lopping off decimals less than 0.5, and rounding up for decimals 
greater than or equal to 0.5.

Another possibility is to use the factorial function. Thus, 4! = 24. 
Different combinations of factorial expressions can be helpful.  

Obviously, the square root function is nice to use as well. 

Depending on the mathematical sophistication of your students, you can 
also incorporate log, ln, and/or trig functions as well. For example, 
[(sin 44 deg. x 4)^4] = 59 where the brackets indicate the greatest 
integer function.

As for how to proceed when playing this game, you could let the 
students decide. In one case, have students continue generating 
numbers, marking the missing ones with blanks. Post these on a 
bulletin board so that they can go back and try to figure out the 
missing combinations. 

Alternatively, create teams of students to play this game. See how far 
each team can go and how many missing numbers are encountered along 
the way. Decide the "winner(s)" by agreeing on the criteria ahead of 
time.

Another interesting challenge is to see how many different ways a 
given number can be generated using the four 4s. For example, 
44/44 = 1 and 44^(4-4) = 1, 4-4+4/4 = 1 etc. 

Finally, another variation of this game is to use the digits in a 
year: for example, 1, 9, 9, and 7 (1997). How many of the counting 
numbers can you generate using any proper math function and these four 
numbers in sequence? For example: 1 * sqrt(9) - (9 - 7)=1 It's the 
same idea.

For examples, see Ruth Carver's puzzles on the Math Forum site:

  http://mathforum.org/k12/k12puzzles/   

This is a great game that will keep your students occupied all year 
and then some. Perhaps on a long journey with their parents!

-Doctor Chita,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/