Phone Number Equations

Here is a new variation to an old numerical puzzle activity. Such puzzles as these are often used to bring a worthwhile computational challenge to the math classroom. This one takes ordinary phone numbers to form equations in a special way. An example will serve to illustrate the basic method.

The author's own phone number happens to be 371-8957. An equation can be formed in the following manner:

3 + 7 + 1 = (8 × 9 + 5) ÷ 7

The exchange portion (i.e. the first three digits) forms the left side of the equation while the remaining four digits constitute the right side. In its purest form, the digits should be left in their natural order. However, for less able students this condition could be waived, and the digits could be combined in any convenient order. For the above phone number, one possibility out of many could be

7 - 3 - 1 = 9 + 7 - 8 - 5

Making phone number equations is not generally an easy task, so the game is not a trivial one. However, in this writer's experience, a little patience usually pays off in the long run for most phone numbers. To demonstrate this, here are several "genuine" phone numbers--of friends of the writer or local businesses, etc.--that have been transformed into PHONE NUMBER EQUATIONS. Note that sometimes two ways may be found for the same number.

```	597-2153	5 + 9 - 7 = 2 + (15 ÷ 3)
59 - 7 = -2 + 1 + 53

339-6733	3 + 39 = 6 × 7 + 3 - 3

965-1626	9 + 6 + 5 = 16 - 2 + 6

747-4578	7 - 4 + 7 = 4 + 5 - 7 + 8

385-0113	3 + 8 - 5 = (0 + 1 + 1) × 3

597-8774	(5 + 9) ÷ 7 = (8 + 7 - 7) ÷ 4
```

This article of mine is copied from The ILLINOIS MATHEMATICS TEACHER, Vol. 29, No. 4, September 1978.

Update (3/13/02)

Here are some great results done by some new friends and math students here in El Salvador. The first group of equations comes from one very clever young man, named Gerardo Zelaya.

2 - 4 + 3 = -15/3 + 6 = 1
(2 + 4)/3 = 1 x (5 + 3) - 6 = 2
2 + 4 - 3 = 1 + 5 + 3 - 6 = 3
2sqrt(4!/3!) = 1 x C(5, 3) - 6 = 4
-2 + 4 + 3 = 15/(-3 + 6) = 5
(-2 + 4) x 3 = 15 - 3 - 6 = 6
2sqrt(4) + 3 = 153 + 6 = 7
2 x 4!/3! = -1 + 5 x 3 - 6 = 8
2 + 4 + 3 = 1 x 5 x 3 - 6 = 9
-2 + 4 x 3 = 1 + 5 x 3 - 6 = 10
2 x 4 + 3 = 15/3 + 6 = 11
2sqrt(4) x 3 = 15 + 3 - 6 = 12
24 - 3 = -1 x 5 + 3 x 6 = 13
2 + 4 x 3 = 1 - 5 + 3 x 6 = 14
-2 + 4! - 3! = -1 + 5 + 3! + 6 = 16
(2 + 4) x 3 = 15 - 3 + 6 = 18
24 - 3 = 1 x 5 x 3 + 6 = 21
2 x 4 x 3 = 15 + 3 + 6 = 24
-2 + 43 = 1 x 5 + 36 = 41
2 + 43 = 15 x (-3 + 6) = 45
24 x 3 = 1 + 53 - 6 = 48
2 x 4! + 3 = 15 + 36 = 51

The next pair of equations were created by Roberto Serrano.

2 x 7 - 8 = sqrt(3 x 72/6) = 6
2 + 7 + 8 = 3 x 7 + 2 - 6 = 17

Here are the equations prepared by Humberto Sermeņo.

-2 + 7 + 4 = 4 + 4 + 9 - 8 = 9
2 - 7 + 4 = 4 - 4 - 9 + 8 = -1
|2 - 7| x 4 = 4(4 + 9 - 8) = 20
2 x 7 x 4 = |4 x 4 - 9 x 8| = 56
2 x 7 + 4 = 4 x 4 x 9 / 8 = 18
2(7 - 4) = 4 x 4 x sqrt(9) / 8 = 6
27 / 4 = 4 x 4sqrt(9) / 8 = 32

The next four come from the mind of Alejandro Quijada.

2(-2+5) = (0)(8) + 6 + 2 = 8
2 - 2 + 5 = 0 + 8 - 6/2 = 5
(2 - 2)(5) = (0)(862) = 0
sqrt(2 + 2 + 5) = (0)(8) + 6/2 = 3

Now we have some equations from Mirna Galdamez.

45 x 1 = 9 x (3 + 3 - 1) = 45
4 + 5 + 1 = 9 - 3 + 3 + 1 = 10
4 + 5 x 1 = 9 - 3 + 3 x 1 = 9
4 + 51 = 9 x (3 + 3) + 1 = 55
4 x 5 + 1 = 9 + 3 x (3 + 1) = 21

Here are Saul Blanco's equations.

3 x 3 - 4 = 1 x (-1 - 2 + 8) = 5
3 + 3 x 4 = -1 x 1 + 2 x 8 = 15
3 + 3 + 4 = 1 x 1 x 2 + 8 = 10
3 - 3 + 4 = -1 - 1 - 2 + 8 = 4

Now Dalia Morales presents her equations. [But what is her phone number?]

2 x 2 x 9 = 8 x 0 + 9 x 4 = 36
22 - 9 = 8 + 9 + 0 - 4 = 13
9 - 2 - 2 = 9 - 8 + 4 + 0 = 5