The teacher called it an opportunity test! Diane half smiled as she
tilted her head back and using her fingers, brushed her long black hair
over and behind her ears. Mr Smiley called it an opportunity test
because it gave everyone a chance to make an A for the year. It was Mr.
Smiley's tradition. Anyone who made an A on his math opportunity test on
the last day of school would earn an A for the year.
Diane intended to make that A. Today she had Explorer with her.
Explorer was Diane's magical friend that had helped her for as long as
she could remember. They could talk silently to each other. Explorer
would make sure she made a perfect grade.
Diane looked at the first problem. Add 1024 + 2048 + 4096
+ 8192 + 16384. Diane asked Explorer "Well, do you know the answer?"
Explorer sounded amused. "Your teacher is making it easy on
himself. This is a doubling series. Look at the first two numbers, 1024
and 2048. What is 1 + 2?"
Diane looked again. "1 + 2 is 3. So what?"
Explorer said "You will see. The third number is 4096. What is 3 + 4?"
Diane said "7. So 1 + 2 + 4 is 7. And 7 is 1 less than 8."
Explorer said "Now you are getting it.
Diane looked at the last two numbers 8192 and 16384. "7 + 8 is 15
and 15 is one less than 16. I see that 1 + 2 + 4 + 8 + 16 is 31 because
31 is 1 less than twice 16."
Explorer said "You have it right. How can you be sure of it?"
Diane had a sudden inspiration. "Suppose it had been 1 + 1 + 2 + 4 + 8
+ 16. Since 1 + 1 is 2, it would have been equal to 2 + 2 + 4 + 8 + 16.
But since 2 + 2 is 4, it would have been equal to 4 + 4 + 8 + 16. But
since 4 + 4 is 8, this would be equal to 8 + 8 + 16. And since 8 + 8 is 16
this would be equal to 16 + 16 = 32. So, If 1 + 1 + 2 + 4 + 8 + 16 is
32, then 1 + 2 + 4 + 8 + 16 must be 1 less than 32 which is 31.
Explorer said "Now you are ready to do the original problem. What is
1024 + 2048 + 4096 + 8192 + 16384?"
Diane saw that 1024 + 1024 was 2048. And 2048 + 2048 was 4096. And
4096 was half of 8192. And 8192 was half of 16384. She said "I see
that this really is a doubling series. If it were 1024 + 1024 +2048 +
4096 + 8192 + 16384 then the answer would be twice 16384. Twice 16384
is equal to 32768. This means that 32768 is 1024 too much. So the
correct sum is 32768 - 1024 which is equal to 31744." Diane wrote
down her answer.
Diane looked at the second question. Add 21 + 42 + 84 + 168 + 336 +
672. "aha", Diane thought, "this is another doubling series." Diane
verified that 21 was half of 42, that 42 was half of 84, that 84 was half
of 168, that 168 was half of 336 and that 336 was half of 672. Diane
thought "If this had been 21 + 21 + 42 + 84 + 168 + 336 + 672 then the
answer would have been twice 672 which is equal to 1344. Therefore the
correct answer is 1344 - 21. Diane wrote down 1323 for her answer to the
second question.
Diane looked at the third question. Add 3 + 6 + 12 + 24 + 48 + 96 +
192 + 384. Diane verified that this was a doubling series as she read
the numbers. She then doubled 384 and subtracted 3. Diane smiled as she
wrote 765 down for the answer to the third question. "This is fun", she
said.
The fourth question was different:
512 - 256 + 128 - 64 + 32 - 16 + 8 - 4 + 2 - 1 = ?
Explorer waited. "You do notice that each number is twice the value
of the following one?"
"So what?" Diane tossed off casually, then she looked again. "Wait,
now I see... 256 from 512 is, well, half, or 256. And for 64 from 128, it's
again half, and so on. This isn't a doubling series, it's a 4-times series!
You can split it into pairs, and each pair is really just half of the first
number." She quickly rewrote the problem:
(512-256) + (128-64) + (32-16) + (8-4) + (2-1) =
256 + 64 + 16 + 4 + 1 = ?
Having found the pattern, Diane was only momentarily stumped. "How
would I add it, hmm."
"How many more..." Explorer began, only to be quickly interrupted.
"You think too fast," chided Diane. "To make the 4, I'd need three more
1's, but then I still need two more 4's to make the 16, so I really need two
more of everything."
Explorer said "Very good Diane. So what is the answer?"
Diane said "when I subtract the 1 from 1024 to get 1023 I still have
every number 3 times. So I have to divide by 3 to get the correct answer."
Diane divided 1023 by 3. Diane smiled as she wrote 341 for the answer to
the fourth question.
Diane looked at the fifth question. Add 729 + 364 + 182 + 91 + 45 +
22 + 11 + 5 + 2 + 1. "Help me Explorer. This is not a doubling series.
What trick is Mr Smiley playing on us now?"
Explorer said "This is a halving series. It is almost a doubling
series in reverse. What would you have to add to the 2 + 1 to make 2 + 2?"
Diane said "I would have to add 1 of course. But 2 + 2 makes only
4, not 5. I'm stuck."
Explorer said "Not really. Just note that you have to add another
1 to the 4 to make 5. Then you can add 5 + 5."
Diane said "But how does that help? 5 + 5 is 10, not 11. Wait.
Now I get it. I just count the times I have to add 1 to get to the next
number. If the next number is odd then I have to add 1 to get to it. If
the next number is even, I am already to it. So the number of times I
have to add 1 is the number of odd numbers in the halving series."
Diane counted the number of odd numbers in the halving series 729 +
364 + 182 + 91 + 45 + 22 + 11 + 5 + 2 + 1. "There are 6 odd numbers in
the halving series. So the answer is 6 less than twice 729." Diane
doubled 729 to get 1458, and then subtracted 6. Diane wrote down 1452 for
the answer to the fifth question.
Explorer said "You have the answer right Diane. But only because
the last number in the halving series was 1. If it had been different
than 1, you would have gotten a wrong answer by your rule."
"So what rule should I have used?" asked Diane.
Explorer said. "The rule should have been that the answer is twice
the first number minus the last number minus the number of odd numbers not
counting the last number."
Diane said "I see. Twice 729 is 1458. The last number is 1. The
number of odd numbers not counting the last number is 5. So the answer
is 1458 - 1 - 5. I get the same answer 1452."
In this way Explorer helped Diane earn an A in her seventh grade
math class for the year.
You too can explore numbers. Make up your own opportunity test and
try to make an A on it.