EXAMPLE 1.13 USING VARIABLES (AMANDA’S TELEPHONE NUMBER EMAIL)
Your friend, Amanda, sends you the email below which she and a number of her friends
have found amazing. Does it always work? If so, why?
“Here is the math trick so unbelievable that it will stump you.
1. Grab a calculator.(You won’t be able to do this one in your head)
2. Key in the first three digits of your phone number (not the area code)
3. Multiply by 80
4. Add 1
5. Multiply by 250
6. Add the last 4 digits of your phone number
7. Add the last 4 digits of your phone number again
8. Subtract 250
9. Divide number by 2
Do you recognize the number?”
Solution Understand the Problem
To make sure that we understand what’s going on, the first step is to see if the procedure
works with our own phone number. Let’s first try Amanda’s phone number of 257-6821.
Table 1: Verification of Amanda’s math trick using her phone number, 257-6821
Step
Result
2. Start with 257
257
3. Multiply by 80
80 (257) = 20,560
4. Add 1
20,561
5. Multiply by 250
(250) (20,561) = 5,140,250
6. Add 6821
5,147,071
7. Add 6821 again
5,153,892
8. Subtract 250
5,153,642
9. Divide by 2
2,576,821
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Step 6 has some ambiguity because the instruction, “add the last four digits” could mean
that we should add 6, 8, 2, and 1 or add the number 6821. If you try the first way, you’ll
see that the process fails and so we took the second interpretation.
The process does work for Amanda’s number because 2,576,821 of Step 9 is 257(10,000)
+ 6821. Thus, when expressed as a phone number, 2,576,821 is 257-6821. Since there is
nothing special about the number used, it may well be that the “math trick” always
works. On the other hand, we can’t try all the phone numbers so just guessing and
checking is not feasible. We should go to the next Polya Principle to see if Amanda’s
trick always works.
Devise a plan
The phrase, “Does it always work?” of the problem gives a direction for a plan. Example
1.12 is solved by looking at all possibilities at once by letting the time needed to
whitewash the fence be the variable T. Let’s see whether that approach will work for
Amanda’s problem. The first step in planning is to decide what variable to use for which
quantity. It is tempting to let x be a phone number. If you try this approach, you will see
that we can’t do steps 2 or 3 in terms of x because we must have only the first three digits
of the seven-digit phone number. Looking at the procedure again, we see that, although
there are 7 digits, only the first three and the last four are used, not any of them
individually. Our plan then is to name the first three digits with the variable y and the last
four with the variable z and go through the Polya Four Principles using algebra.
Carry Out the Plan
Let the telephone number be given and y be the first three digits of the number and z the
last four. (Thus in the case of Amanda’s number, 257-6821, y = 257 and z = 6821.). We
will now follow Steps 2 through 9 exactly as in Table 1 and see what happens.
Table 2: Verification of Amanda’s math trick using the phone number whose first 3
digits are y and last four are z.
Step
Results
2. Start with
y
3. Multiply by 80
80y
4. Add 1
80y + 1
2
5. Multiply by 250
250(80y + 1) = 20,000y + 250
6. Add z
20,000y + 250 + z
7. Add z again
20,000y + 250 + 2z
8. Subtract 250
20,000y + 2z
9. Divide by 2
(20,000y + 2z) / 2 = 10,000y + z
What is the final answer after all 9 steps? The result at step 9 is 10,000y + z. However,
10,000y takes the 3 digit number y (257 in our example) and adds four zeros (to get
2,570,000). Thus the first 3 digits of the 7 digit number 10,000y + z are the digits of the
number y.
What are the last 4 digits of 10,000y+ z? They are exactly the digits of z (for example,
2,570,000 + 6821 = 2,576,821). The “math trick” works because 10,000y + z is a way to
express in algebra the phone number (without area code) y - z (that’s y dash z, not
subtraction). Note that we didn’t need a calculator to do the problem!
Look Back
In the explanation of Amanda’s phone number problem, we used “Algebra as the Great
Explainer” because we were trying to show that something involving numbers was
always correct. At first, we thought of solving the problem using only one variable but
recognized that there were really two different numbers involved; that is, a phone number
is made up of two parts (plus an area code) and we needed different names for the two of
them. We then went through some algebraic steps and finished the problem. Since we
proved the result for all seven-digit numbers, not just some specific ones, we have given
a carefully reasoned argument that shows the result is always true. This approach is also
an example of the “Proof and Reasoning Standard” of the NCTM.
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