Locker problem

Navajo Nation Math Circles Project http://navajomath.wordpress.com/
Locker Problems - July 14, 2015
by Amanda Serenevy, Riverbend Community Math Center, South Bend, IN & Bob Klein, Ohio University, Athens, OH
http://www.riverbendmath.org/ & http://about.me/kleinbob/
Note: These materials were culled from a number of sources including Elgin Johnston (Iowa State U.) and Mark
Brown (MidAmerica Nazarene U.
The session was 1.5-hours long and there were 15 students. We began by having students go outside and trying to
enact the classical locker problem. At first we had students face the wall to represent closed lockers and turn around
when “opened” by a student tapping them on the shoulder. This didn’t work so well because students weren’t able
to see what was going on. So in the second group of students we tried having students gather in a semi-circle and
represent ‘open’ with an open hand held in front of them and ‘closed’ with closed hands at their sides. Students quickly
got lost because they were to fidgity to hold a hand position. Perhaps having students have two-colored signs, one side
saying “#1 OPEN” and one saying “#1 CLOSED” (for locker number 1) would be the better approach and allow
students to act the problem out while staying organized and also seeing the emerging patterns.
Analyzing the problem as we went bogged down the rhythm and distracted from the task. As such, it’s best to simply
act out the problem and save analysis for later.
Upon returning to the classroom inside, students were given bags of two-color counters and the following
problem:
Suppose we had 10,000 closed lockers. We elect person #1 to open every locker numbered a multiple of 1
(hence all). Person #2 changes the state of all lockers numbered a multiple of 2 (opening closed lockers and
closing open lockers). Person #3 changes the state of all lockers numbered a multiple of 3 (and so on). If
all 10,000 people have a chance to do this, (at this point we asked students to generate interesting mathematical
questions, leading to: Which lockers remain open and which closed?
Students investigated with at least 20 two-color counters each, discussing with each other, writing down
conjectures, etc., for around 20 minutes. Students gradually saw that lockers {1, 4, 9, 16} remained open.
We pushed them to determine “What is the next number?” to try to get them to generalize the pattern.
Students generally came to see the pattern one of three ways:
1. (Odd Differences: most common). Students calculated the differences of successive numbers to see
the odd numbers emerging. That is: 4 - 1 = 3; 9 - 4 = 5; 16 - 9 = 7. At this point, they guessed that the
next number should differ from 16 by 9, thus it should be 16 + 9 = 25.
2. (Squares). Students recognized {1, 4, 9, 16} as the first four square numbers ({12, 22, 32, 42}) and con-
jectured that the next number should be 52 = 25.
3. (Even Spacing: least common). This is similar to #1. above, but students here saw that there were
two numbers between 1 and 4, four numbers between 4 and 9, six numbers between 9 and 16, so
conjectured that there would be eight numbers between 16 and the next open locker number of 25.
All of the options above were presented by students to the class. At this point, a new problem was added
to the initial problem posed (“Which lockers are open/closed?”), namely, “Why?” We emphasized that
experimenting and observing led to answers to the first problem, but understanding why this takes place
is the real work of mathematicians, scientists, and critical thinkers generally.
In particular, we explored the following:
• How were the three descriptions of the patterns (above) related to each other?
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• Why are the square-numbered lockers open?
How were the three descriptions of patterns related?
Students quickly recognized how items 1 and 3 were related.
Relating the first and third patterns was a matter of establishing that successive squares always differed
by successive odd numbers. This was motivated by having students gather their two-color counters into
square arrays. Four could be represented as 22 or a 2 × 2 square as below.
Students worked a while in paired-discussions to see that we must add five dots to the “right and bottom
edges” of the 2 × 2 square to get a 3 × 3 (nine) square as below. The five diamond-shaped points outside of
the dashed lines form a 3 × 3 square from a 2 × 2 square.
Students confirmed the pattern for moving from a 3 × 3 to a 4 × 4 square and a 4 × 4 square to a 5 × 5
square. They were convinced that the pattern held.
At this point, one of the facilitators announced that he saw that it was true for those numbers, but was
unsure about higher numbers. Would it work for moving from a 755 × 755 square to a 756 × 756 square?
If so, how many dots were we adding? Students seemed surprised that the facilitator would doubt the
pattern that was obvious to them. This was an opportunity to emphasize that showing that some examples
work does not prove the statement generally.
This was our springboard to prove it algebraically. We asked students to examine the case of moving from
a 5 × 5 square to a 6 × 6 square. Using a picture similar to that below, we asked students, “What did you
do in going from the 5 × 5 to the 6 × 6 squares? How did you add the dots?
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Students expressed this in two ways though we had to push them from “We added 11 dots” to a more
explicit explanation of how they added those dots. We had them explain it relative to the picture.
1. “We started with a 5 × 5 square then added 6 dots on the right side (column) followed by an additional
5 dots along the bottom edge (row) to make a 6 × 6 square.” As they described this, we wrote 5 × 5 +
6 + 5 = 6 × 6 on the board and asked the students if (i) it represented the method that was described,
and (ii) if the equation was true. This description is represented by the picture below.
5 × 5 = 25 6
5
Picture representing 5 × 5 + 6 + 5 = 6 × 6.
2. “We started with a 5 × 5 square then added ﬁve dots in the right column, ﬁve dots in the bottom row,
then 1 dot to make it a square.” We wrote on the board 5 × 5 + 5 + 5 + 1 = 6 × 6 and asked questions
(i.) and (ii.) as above. This approach is represented by the picture below.
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5 × 5 = 25
5
5 1
Picture representing 5 × 5 + 5 + 5 + 1 = 6 × 6.
At this point, we asked a student to come to the board and pick their favorite letter to represent the size
of the square (here we will use k). We then used the representation of their explanations for the 5-to-6 case
to help them convert, in the case of the ﬁrst approach, 5 × 5 + 6 + 5 = 6 × 6 into k × k + (k + 1) + k
?
=
(k + 1)(k + 1) (or (k + 1)2). Having the students use the
?
= symbol is key to emphasizing that we have not
proven this statement to be true yet–it’s equality is still in question. In the case of the second approach, we
help students convert 5 × 5 + 5 + 5 + 1 = 6 × 6 into k × k + k + k + 1
?
= (k + 1)(k + 1) (or (k + 1)2).
Again, it is important to get students to see that writing down an equation doesn’t prove the equation true.
For that, we must use algebra and make the left-hand side (LHS) look like the right-hand side (RHS). We
divided the group into the LHS of the room and the RHS of the room and had each focus on simplifying
(LHS) or expanding (RHS) the terms to make one look like the other. In some cases, we used the idea that
(k + 1)2 could be represented as a square of side-lengths k + 1 and divided the square as below, calcultating
the areas of each piece and recognizing that the sum of the areas of the pieces must equal the area of the
whole.
k
k2
k 11
1
k k
Picture showing how (k + 1)2 = k2 + k + k + 1.
At this point, the students insisted that showing RHS = LHS established the result generally and the facili-
tator indicated that he was satisﬁed.
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Why are only the square-numbered lockers open?
In answering this question, students may need to be reminded of the rules of how the lockers were open
and closed. One way to approach this is to ask about a few speciﬁc lockers. For instance:
• “Who touched locker 15?” We write on the board: 1 opened it, 3 closed it, 5 opened it, 15 closed it.
• “Who touched locker 7?” We write on the board: 1 opened it, 7 closed it.
Invite students to the board to show a few more cases including a square. Students need time to connect that
the rules of opening and closing lockers involve things like “all multiples of 3” so it’s natural to consider
the factors of locker numbers as well. They should see from the board examples that only the factors of
each number touch that numbered locker.
Prod students to explore how the squares’ factors differ from the non-squares’ factors. They should see
eventually that non-square numbers have factor lists that can be paired so that, for instance, 15 has pairs
1 × 15 and 3 × 5 whereas square numbers always end up pairing one factor with itself, as in 9 which pairs
1 × 9 and 3 × 3. Hence for locker number 9, it’s being touched an odd number of times so that if it began
closed, it should end open. In the case of non-square numbers, because the factors come in pairs, there will
be an even number of factors indicating pairs of open-close open-close ... so that if it starts closed, it should
end closed as well.
Extensions:
Given the wide range of backgrounds in our session, some students needed extensions to explore while the
rest of the students caught up to the problems stated on the board. The following questions were interesting
to students and provided sufﬁcient challenge while being relevant to the discussion.
3. Of the 10,000 lockers, how many remain open at the end?
4. What if only the even numbered people went down the line changing the states of the lockers (that
is, 1 does nothing, 2 changes all multiples of 2, 3 does nothing, 4 changes all multiples of 4)? What
would remain open then?
5. What if only the odd numbered people went down the line changing the states of lockers?
6. What if only the prime-numbered people went?
7. What if person number 3 is ill and has to skip her turn? If everyone else went down the line, which
lockers are open at the end?
8. What if person number 3 AND person number 9 are ill and have to skip their turns?
9. If we want to end up with locker number 1 open and all others close, whom should we send down
the line?
10. What if we only want locker number 3 open?
11. What if we only want the prime-numbered lockers open at the end? Whom should we send down the
line?
12. What questions could you invent?
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