Sandwiches

522 /www/nrich/html/content/97/09/six1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <div style="float: left;"><mdo:image height="80" width="195" src="oneone.jpg" alt="1, 1, 2, 2, 3, 3"></mdo:image></div> <div>Suppose you have two 1s, two 2s and two 3s. Arrange these six digits in a list so that:</div> <div>between the two 1s there is one digit, giving 1?1,</div> <div>between the two 2s there are two digits, giving 2??2, and</div> <div>between the two 3s there are three digits, giving 3???3.</div> Can you do the same if you only have 1s and 2s? Explain your answer. Can you do the same if you include two 4s, and between the two 4s there are four digits? Here is a solution using 5's, 6's and 7's as well: <mdo:image height="56" width="378" src="7sandwich.jpg" alt="7 sandwich"></mdo:image> Find other solutions with all these digits. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Thank you Tim from Gravesend Grammar School for your solution. The solutions are 312132 and its mirror image 231213. We call these 3-sandwiches. You cannot do the same with only ones and twos, because between the two twos there must be two digits, which have to be ones as these are the only available digits, but that means that between the two ones there are no digits, so this is not possible. The only 4-sandwiches are 41312432 (and its mirror image). You will probably have found quite a lot of 7-sandwiches because there are altogether 26 (and their mirror images). You may have tried, and failed, to find 5-sandwiches and 6-sandwiches but there aren't any. If you have tried to find 8-sandwiches you will probably have had more success as there are altogether 150 of these (and their mirror images). See <a href="http://nrich.maths.org/public/viewer.php?obj_id=1364&part=index"> 'Impossible Sandwiches'</a> for a simple proof that n-sandwiches can only be made when n=4m or n=4m-1 (for integers n and m). For example solutions do not exist for n=5 or n=6 but solutions do exist for n=7 and 8. The following BASIC program gives the 4-sandwiches. You could try making some small changes to get the 7-sandwiches. Alternatively you could take up the challenge of finding all the solutions without using a computer, or write a computer program in a different language to find the solutions. <pre> 10 DIM A(8) 20 FOR A(1)= 1 TO 6: REM A(1) and A(5) are positions for 1's 30 FOR A(2)= 1 TO 5: REM A(2) and A(6) are positions for 2's 40 FOR A(3)= 1 TO 4: REM A(3) and A(7) are positions for 3's 50 FOR A(4)= 1 TO 3 REM A(4) and A(8) are positions for 4's 60 FOR T=1 TO 4 70 A(T+4)= A(T) + T + 1: REM Second positions of 1 to 4 spaced correctly from the first 80 NEXT 90 PROCCHECK 100 IF Z = 1 PROCPRINT 110 NEXT: NEXT: NEXT: NEXT 120 END 130 DEF PROCCHECK 140 Z=1: REM Z=0 eliminates cases where two numbers occur in same position 150 FOR I=1 TO 7 160 FOR J=I+1 TO 8 170 IF Z=0 GOTO 200 180 IF A(I)=A(J) THEN Z=0 190 NEXT: NEXT 200 ENDPROC 210 DEF PROCPRINT 220 FOR M=1 TO 8 230 FOR K=1 TO 8 240 IF A(K)=M AND K < 5 PRINT K; 250 IF A(K)=M AND K > 4 PRINT K- 4; 260 NEXT: NEXT 270 PRINT 280 ENDPROC </pre> There are no N-sandwiches for N=1, 2, 5, 6 or 9 or any number that leaves a remainder of 1 or 2 when divided by 4. (Click <a href="/content/97/09/six1/proof.html" onclick="mediaSave()">here</a> for a proof of this fact.) </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Why do this problem? <a href="http://nrich.maths.org/public/viewer.php?obj_id=522&part=index">The Sandwiches problem</a> offers a challenge for everyone at all levels. It is particularly valuable in primary schools because it gives young learners an opportunity to work out a proof and to explain it. They can find the solution for the 3-sandwich (consisting of 1's, 2's and 3's) and discover that it is impossible for the 2-sandwich. Even very young learners can explain why (prove that) it is impossible to find a solution for the 2-sandwich. You may decide to stop at 3-sandwiches or try to find 4-sandwiches as well. At upper primary and lower secondary level, as there are many solutions in the case of 7-sandwiches and 8-sandwiches, the problem provides an opportunity for many individual learners to have success in discovering their very own solution, different to any that have already been found. Having investigated 2, 3, 4 and 7-sandwiches, if learners in your class have been encouraged to ask "What if..." and look for generalisations, the natural question is "What about 5-sandwiches and 6 sandwiches?" The experience of learning to think mathematically offered by this problem is equally valuable to learners in upper secondary school. Possible approach It is helpful, particularly for young learners, to have digits to rearrange (either plastic or simply written on paper or card). It is a good idea to have a 'Challenge-Chart' on the classroom wall where new solutions can be written up as people discover them. As there are many solutions for n=7, this problem calls for you to work systematically in order to find them all. Key question What sandwiches can you make? Can you make 2-sandwiches and if not why not? Are any sandwiches the same looked at in different ways? Is it possible to make 5-sandwiches? For which values of n can n-sandwiches be made and for which values of n is it impossible? Why? Possible support You may want to stop with n=4. Possible extension The process of working systematically through all possible cases can be tedious and time consuming. This is where discussion by the class of the best systematic approach, and then sharing the work of searching for solutions and checking, offers a valuable learning experience. Such work prepares learners for using computers in problem solving and not just in mathematics but also in other fields. In order to construct a computer program you have first to plan a systematic approach to the problem. Another extension could be to write a computer program to find solutions and to test which values of n yield n-sandwiches. Finally there is a method of proving that for certain values of n it is impossible to make n-sandwiches. The argument is simple and requires only very elementary mathematics. The proof falls into the 'Aha' category - once you see it, it seems obvious and amazingly simple, but the choice of method called for real inspiration in the first place. This is an existence proof not a construction proof - it does not give a method of constructing solutions. See the article <a href="http://nrich.maths.org/public/viewer.php?obj_id=1364&part=index">Impossible Sandwiches.</a> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> It is easy to find some solutions. Can you be sure you have found them all? Start with 1, 1, 2, 2, 3 and 3. Then experiment until you find a solution. Now try 1, 1, 2 and 2. Is it possible? If not, why not? What about 1, 1, 2, 2, 3, 3, 4 and 4? </mdoxml> 2 5 0 1 1 1 1 Sandwiches Arrange the digits 1, 1, 2, 2, 3 and 3 so that between the two 1's there is one digit, between the two 2's there are two digits, and between the two 3's there are three digits. Using, Applying and Reasoning about Mathematics Trial and improvement Using, Applying and Reasoning about Mathematics Working systematically