6282
/www/nrich/html/content/id/6282/
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2011-04-27T10:28:22
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This challenge is based on the game <a href="http://nrich.maths.org/1246&part=">Frogs</a> which you
may have seen before. <br></br>
<br></br>
There are two green frogs and two brown toads:<br></br>
<br></br>
<mdo:image width="452" height="92" alt="" src="TwoFrogs.png"></mdo:image><br></br>
<br></br>
A frog or toad can jump over one other creature onto an empty
lilypad or it can slide onto an empty lilypad which is immediately
next to it. <br></br>
Only one creature, at a time, is allowed on each lilypad.<br></br>
<br></br>
Now the idea is for the frogs and toads to change places. So,
the frogs will end up on the side where the toads started and
the toads will end up where the frogs began.<br></br>
<br></br>
The challenge is to do this in as few slides and jumps as
possible. <br></br>
<br></br>
You could use the interactivity below to help you try out your
ideas.<br></br>
<br></br>
To move a frog or toad, click on it. If it is not able to
jump or slide, it won't move. The toads and frogs in the
interactivity only move in the direction they are facing.<br></br>
<br></br>
This button <mdo:image width="26" height="26" src="Button1.jpg" alt="button 1"></mdo:image>takes you back
to the start.<br></br>
<br></br>
This button<mdo:image width="26" height="26" alt="button 2" src="Button%202.jpg"></mdo:image>takes you back one step at
a time.<br></br>
<br></br>
This <mdo:image width="108" height="82" alt="score" src="Score.jpg"></mdo:image>keeps a record for you - you may
also be interested in the grand total of all four parts added
together.<br></br>
<br></br>
<a href="/content/id/6282/frogs.swf">Full screen
version</a> <br></br>
<mdo:flash height="400" width="500"><param value="/content/id/6282/frogs.swf" name="movie" ></param><param value="7" name="flashplayerversion" ></param></mdo:flash><br></br>
<br></br>
<br></br>
How do you know you have found the smallest possible number of
slides and jumps?<br></br>
<br></br>
Why not try three frogs and three toads? <br></br>
What is the smallest number of slides and jumps now?<br></br></mdoxml>
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<div>
<p><span class="editorial">Towards the end of August we had some last minute solutions sent in from different countries, namely - England, New Zealand, Australia, China and Denmark. We had a very precise, accurate and well-explained solution sent in earlier, by Hannah from Leicester High School for Girls in the UK. This represents the most exhaustive solution that I have come across for my
particular activities. Well done Hannah! You have good reason to be proud of your work. I hope you will continue to work on some of the activities from this site.</span></p>
<p> </p>
<p><em>Two Frogs and Two Toads</em> </p>
<p>To keep this simple, let's assign a letter to each of our adventurous amphibians: from left to right, their names are A, B, C and D. At the start, you are able to move any of the $4$ amphibians. It doesn't matter whether you move a toad or a frog first, because the layout is symmetrical. Let's say we move a frog first. Imagine a situation where A jumps over B , we'd be in a right mess because
no amphibian would be able to move anymore. So this means we have to slide B onto the lily pad directly beside it. Now, we can either move A, which would bring us back to the situation mentioned at the beginning, or we could, more sensibly, make C jump over B. Just to recap, our current position is A, C, B, (space), D.</p>
<p>At this point we can either move B or D. We'll have to think ahead. Say we move B onto the vacant lily pad. Now we could move either A or D. However, both these moves would bring us to a dead end and a serious congestion. This means we will have to slide D onto the lily pad beside it. From here, the only thing we can do is make B jump over D. Now we can move either A or D. Moving D would bring
us to a congestion that just won't do, so we are obliged to jump A over C. There is only one move we can make at this point: sliding C onto the left-most lily pad. After this, there still is only one possible move: jumping D over A. From here on it's quite obvious: just slide A nicely in beside B, and we've exchanged the positions of the frogs and toads.</p>
<p>This is not the only solution: the exact reflection of what I've just explained, where the toad moves first, is the other answer. Altogether, this has taken $8$ moves in total, according to the NRICH interactivity. Each amphibiotic species made exactly two slides and two jumps. I can be sure that this is the lowest possible number of moves because I have worked this out
systematically. However, I have found that solving this puzzle in more than $8$ moves is impossible; from several trials I have discovered that doing it in $8$ moves is the only solution. This means that $8$ moves is also the highest as well as lowest possible number of moves for two frogs and two toads. </p>
<p><em>Three Frogs and Three Toads</em></p>
<p>Once again, let's give each amphibian a letter: from left to right, A, B, and C, which are the frogs, and D, E, and F, which are toads. To keep things consistent, I'll start with frogs again, although, again, the puzzle layout is symmetrical. The first part of this puzzle is similar to the previous one, but I'll explain that bit again anyway; firstly, we want to move one of
the frogs, so that's either B or C. B will bring us into a mess, so we have to move C. Moving B at this point won't be a good idea, and neither will moving A, so we'll settle with jumping D over C. Sliding C onto the empty lily pad will bring a congestion later on so the best idea is to move E. If we slide F, it won't do any good, so let's make C hop over E. Moving E one lily pad along will lead
to congestion so B should jump over D. If we move D over, it will lead to trouble, so we should move A onto the vacant lily pad. We now have an alternating pattern of frogs and toads, which is a good sign. From here it should be quite straightforward; the only thing we can do now is jump D over A. Moving A forward will lead to congestion (I apologise that I keep using that word, but it suits the
situation best!) so I would jump E over B. In a similar pattern, let's jump F over C, and then move C onto the empty lily pad. After that we should jump B over F, and then jump A into the space that B just left. Now we should slide E in beside D, jump F over A and then move A into the spare slot.</p>
<p>This time the outcome is quite different: the interactivity states that frogs made four slides and four jumps, and that toads made two slides and five jumps. This means that toads made fewer moves than frogs. The total number of moves is $15$. I tried this again starting with the toads, and this time the frogs made two slides and five jumps, and the toads made four of each. I can be assured
that this is the shortest possible number of moves because I worked it out systematically and always chose the one move that would work. I think the key is to make an alternating pattern of frogs and toads, and from then on it's easy. </p>
<p>I worked out the smallest number of moves for one frog and one toad, as well as four frogs and four toads, to see if there was a pattern and whether I could come up with a formula.</p>
<p>Here are my results: </p>
<p>$1$ of each type of amphibian, $3$ moves</p>
<p>$2$ of each type of amphibian, $8$ moves</p>
<p>$3$ of each type of amphibian, $15$ moves</p>
<p>$4$ of each type of amphibian, $24$ moves.</p>
<p>I noticed that the number of moves is the number of each type of amphibian multiplied by itself plus 2. Now, that's a bit of a mouthful, so here's the formula that I came up with: $n(n + 2)$ or $n˛ + 2n$.</p>
<p> </p>
<p><span class="editorial">This might encourage others to write explanations in their own words, describing what they have done and where they got to.</span></p>
<p> </p>
</div>
<p> </p>
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<h3><span style="font-weight: bold;">Why do this
problem?</span></h3>
<div><a href="http://nrich.maths.org/6282&part=">This game</a>
allows pupils to think strategically in an engaging context. They
will need to work very systematically and may also want to develop
their own recording system. With very young pupils it helps
to re-inforce the following of rules.</div>
<br></br>
<h3>Possible approach</h3>
<div>You could introduce this challenge using the interactivity,
but it works just as well to have pupils replacing the frogs and
toads. The four chosen children can be sat on chairs with the
rest of the group offering ideas. Having a go at the initial
challenge as a whole class to begin with will help reinforce the
rules and may also bring about the need for some sort of
recording.</div>
<div> </div>
<div>Having got the idea, learners could work in pairs or small
groups. At this stage, depending on your focus, you may offer
them the interactivity, or some pupils will prefer to have a
physical representation in front of them in the manner of small
counters, blocks etc. to move around. Keep a watch out for pupils
who don't have set places or items representing the lilypads, as it
is easy to lose the empty place!</div>
<div> </div>
<div>Emphasise that you are looking out for those pairs/groups who
are able to justify their thinking and convince everyone that there
really isn't a way of doing it in fewer moves. </div>
<br></br>
<h3>Key questions</h3>
<div>Tell me about what you are thinking.</div>
<div>Why that move?</div>
<div>You seem to have some system going on can you tell me about
it?</div>
<div> </div>
<br></br>
<h3>Possible extension</h3>
<div>Some learners will be keen to try larger numbers of frogs and
toads. Being able to predict the total number of slides and
jumps needed for a given number of frogs/toads is not
straightforward but there is still value in encouraging pupils to
convince you there is no quicker way to complete the
challenge.</div>
<div> </div>
<div>Some children may be intrigued by the Towers of Hanoi problem,
which is similar in the necessity to work systematically.
There are three pegs, and on the first peg is a stack of discs of
different sizes, arranged in order of descending size. The object
of the game is to move all of the discs to another peg. However,
only one disc can be moved at a time, and a disc cannot be placed
on top of a smaller disc. The <a href="http://nrich.maths.org/6690&part=">interactivity in
this problem</a> might help.</div>
<br></br>
<h3>Possible support</h3>
<div>Some pupils may need reminding of the rules but the
interactivity may help in this respect.</div>
<br></br></mdoxml>
<mdoxml version="1.0">You could use counters or blocks to represent the frogs and toads
if you're not using the interactivity.<br></br>
How will you keep track of the moves you've made?<br></br>
Where <span style="font-style: italic;">could</span> you move that
frog/toad? What would happen next? So which move might
be better/best?<br></br></mdoxml>
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Junior Frogs
Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?
Using, Applying and Reasoning about Mathematics
Working systematically
Using, Applying and Reasoning about Mathematics
Recording mathematics
Using, Applying and Reasoning about Mathematics
Trial and improvement
Information and Communications Technology
Interactivities