6071
/www/nrich/html/content/id/6071/
1
2011-02-01T00:00:01
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This is a game for two players. You can use the interactivity below, or you could print off a page of blank clock faces in <a href="/content/id/6071/blankclocks.doc">Word</a> or as a <a href="/content/id/6071/blankclocks.pdf">pdf</a>.<br></br>
<br></br>
<br></br>
Set the time on the clock to $6$ o'clock to start the game.<br></br>
Decide who will go first (player $1$) and who will go second (player $2$).<br></br>
Take it in turns to choose to move the hands of the clock on by $\frac{1}{2}$ hour or by $1$ hour. For example, player $1$ could choose $\frac{1}{2}$ hour, so the clock hands move to 6.30, then player $2$ might choose $1$ hour, moving the clock hands to 7.30 ... etc.<br></br>
The winner is the player who moves the hands exactly onto $12$ o'clock.<br></br>
<br></br>
Can you work out a winning strategy so that you can always beat your opponent?<br></br>
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<br></br>
<a href="/content/id/6071/Clock.swf">Full screen version</a><br></br>
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We describe the winning algorithm for the game where moves of only
1/2 hour or 1 hour are allowed.<br></br>
<br></br>
A number of you pointed out that a key time was 10.30 - if we land
on 10.30 then our opponent must either move to 11.00 or 11.30, and
either way we've won. So landing on 10.30 means we're guaranteed to
land on 12.00. <br></br>
<br></br>
The idea is to backtrack, and list a number of key steps around the
clock. Landing on 9.00 means we're guaranteed to land on 10.30 if
we want to, using the same argument above. Similary landing on 7.30
guaranteeslanding on 9.00, landing on 6.00 guarantees landing on
7.30 etc... the full sequence of key times is<br></br>
<br></br>
1.30, 3.00, 4.30, 6.00, 7.30, 9.00, 10.30<br></br>
<br></br>
Any player who lands on one of these times will win for sure, since
they will always be able to move onto the next key time, whatever
the other player does, until they get to 10.30.<br></br>
<br></br>
It follows that if player 1 starts at 3.00, 6.00 or 9.00 then
player 2 can force him to lose, but if he starts on any other time,
then on his first turn he can move onto a key point and make sure
he wins.<br></br>
<br></br>
See if you can now extend this idea to the full game, where players
can move at 15 minute intervals.<br></br>
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<h2>Stop the Clock</h2>
<br></br>
This is a game for two players. You can use the interactivity below, or you could print off a page of blank clock faces in <a href="/content/id/6071/blankclocks.doc">Word</a> or as a <a href="/content/id/6071/blankclocks.pdf">pdf</a> .<br></br>
<br></br>
<br></br>
Set the time on the clock to $6$ o'clock to start the game.<br></br>
Decide who will go first (player $1$) and who will go second (player $2$).<br></br>
Take it in turns to choose to move the hands of the clock on by $\frac{1}{2}$ hour or by $1$ hour. For example, player $1$ could choose $\frac{1}{2}$ hour, so the clock hands move to 6.30, then player $2$ might choose $1$ hour, moving the clock hands to 7.30 ... etc.<br></br>
The winner is the player who moves the hands exactly onto $12$ o'clock.<br></br>
<br></br>
Can you work out a winning strategy so that you can always beat your opponent?<br></br>
<br></br>
<br></br>
<a href="/content/id/6071/Clock.swf">Full screen version</a><br></br>
<br></br>
<br></br>
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</div>
<br></br>
<h3><span style="font-weight: bold;">Why play this game?</span></h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=6071&part=index">Stop the Clock</a> is a motivating context in which children can develop their fluency with telling the time and calculating time intervals. However, the real challenge here is to find a winning strategy and, at the highest level, to be able to beat an opponent whatever the start time and whatever the time
intervals.</div>
<br></br>
<h3>Possible approach</h3>
<div>Introduce the game to the class by playing as a whole group, perhaps one half against the other, several times. Then suggest that children play in pairs, either at computers, or by using sheets of blank clocks (<a href="/content/id/6071/blankclocks.doc">Word document</a> or <a href="/content/id/6071/blankclocks.pdf">pdf</a>) to record their game. Challenge them to find a strategy for beating
their partner.</div>
<br></br>
<div>As they play, circulate around the classroom and ask them what they think is important so far. Some might suggest that in order to win, they must make the clock show 10.30. Others may have thought further back and have ideas about how they can make sure they get to 10.30, and therefore 12.00. After a suitable length of time bring the whole class together and invite one pair to demonstrate
their strategy, explaining their decisions as they go along. Use other ideas from the group to refine the strategy.</div>
<br></br>
<div>You could then choose some extension ideas (see below) for pairs to work on - perhaps different pairs working on a different set-up. Investigating this game fully could become a long-term challenge for the class which you come back to at various stages throughout a term, for example.</div>
<br></br>
<h3>Key questions</h3>
<div>What happens if your opponent gets to 10.30? Why?</div>
<div>How can you make sure you get exactly on $12$ o'clock? What time would you leave on the clock so that you can get to 12.00 on your next go, after your opponent has been?</div>
<div>So, what time would you want the clock to say on the go before that?</div>
<div>How can you work out these "key times" that you must "land on" on your way to the target?</div>
<br></br>
<h3>Possible extension</h3>
<div>Invite children to investigate how their strategy would change if they could choose a different starting time.</div>
<div>What about if they were only allowed to choose quarter of an hour, half an hour or three quarters of an hour?</div>
<div>How would their strategy change if the could choose quarter of an hour or half an hour only?</div>
<br></br>
<h3>Possible support</h3>
<div>The length of the game can be reduced by choosing $9$ o'clock as the start time. A game starting at 9.00 involves the same thinking, which is the important point, but might be more manageable for some children. If pupils are encouraged to record the times that are made and the intervals chosen, then it will help them to notice patterns.</div>
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You could print off a sheet of blank clocks (<a href="/content/id/6071/blankclocks.doc">Word document</a> or <a href="/content/id/6071/blankclocks.pdf">pdf</a>) to record the times you and your partner choose.<br></br>
Play the game several times. What do you notice?<br></br>
What happens if your opponent gets to 10.30? Why?<br></br>
How can you make sure you get exactly on $12$ o'clock? What time would you leave on the clock so that you can get to 12.00 on your next go, after your opponent has been?<br></br>
<div style="clear: both;">So, what time would you want the clock to say on the go before that?<br></br>
<div style="clear: both;">How can you work out these "key times" that you must "land on" on your way to the target?</div>
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Stop the Clock
This is a game for two players. Can you find out how to be the
first to get to 12 o'clock?
Measures and Mensuration
Time
Using, Applying and Reasoning about Mathematics
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Using, Applying and Reasoning about Mathematics
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