Mathematical Challenge 1998-99

Session 1998-99 saw an administrative rearrangement of the Mathematical Challenge competition with five main sections instead of four. Sections 1 and 2 remain largely as they were before, but Section 3 has become slightly smaller by losing Dumfries and Galloway to the new Section 5. Section 5 has also taken over the southern part of the old Section 4. Section 4, still by far the largest section (in terms of number of entries), is split into a primary sub-section and a secondary sub-section. A report from each of the sections follows this general report.

The sections have adopted briefer, more manageable names and their approximate geographical locations can be seen from the following map of Scotland.

A further rearrangement of the competition is taking place in 1999-2000 with the secondary competition moving to two sets of five problems instead of three sets of four. This is partly to ease the administrative burden on both the Challenge organisers and on the teachers in the schools who kindly distribute and collect in the questions. It also recognises that secondary school pupils have many other calls on their time in the second half of the school year, usually resulting in a big drop in the number of entries to Problem Set III. The primary competition will continue to have three sets of three problems.

The sections have adopted briefer, more manageable names and their approximate geographical locations can be seen from the following map of Scotland.

A further rearrangement of the competition is taking place in 1999-2000 with the secondary competition moving to two sets of five problems instead of three sets of four. This is partly to ease the administrative burden on both the Challenge organisers and on the teachers in the schools who kindly distribute and collect in the questions. It also recognises that secondary school pupils have many other calls on their time in the second half of the school year, usually resulting in a big drop in the number of entries to Problem Set III. The primary competition will continue to have three sets of three problems.

The numbers of entries appear to be up in the Senior Division in most sections, but Senior (S5 and S6) numbers are still low. The number of Primary entries continues at the high level of the last two years, although the overall number of Secondary entries has dropped back more towards its level of the mid 1990s. The quality of entries this year was generally high. An exceptionally pleasing solution to one of the Senior problems by a pupil of Boroughmuir High School, Edinburgh, is included after the section reports.

In reading the North Section report, readers may be dismayed to read Professor Patterson, for many years now the National Chairman of the Challenge, talking about his 'last report as organiser for the North Section'. Fear not - although he is retiring as North Section Organiser, he is continuing as National Chairman for a while at least. During the 1998-99 session, the National Committee welcomed various new members Hannah Fulford of Invergordon Academy has joined the National Committee. Frank Smith of St. Andrews University took over as Organiser for the East & Central Section and Elizabeth West of Paisley University has organised the new South West Section. Gerry McKaig of St.Andrews College, already a much valued National Committee member, became Organiser of the West Primary Section. At the end of the session, the Committee bade farewell and expressed their gratitude to David Weir, who had kindly stepped in to keep Section 4 afloat, and to Judy Goldfinch, the Deputy Chairman, whose term of office on the SMC had come to an end.

Early in the session, the Scottish International Education Trust and The Edinburgh Mathematical Society both agreed to renew their support for Mathematical Challenge for three-year periods. A successful application was made to the London Mathematical Society for a grant for 1998-99 and a kind offer from the Society of Petroleum Engineers proved to be very helpful. Professor L E Frankel of the University of Bath made another much appreciated donation. Other local sponsors are mentioned in the section reports. The appreciative comments made by Professor Patterson in his report below for the North Section, about the invaluable contribution of the markers, the question setters, and also of the teachers in the schools, applies to all sections - we really do appreciate your help: in fact, the competition could not continue without such support. Thank you all.

J M Goldfinch

Section 1: North

This is my last report as organiser for the North Section, which until very recently has always been known as Section 1. I am very pleased to say that Mathematical Challenge North Section continued to flourish in session 1998-99 and that it is now in the capable hands of Dr. Colin Maclachlan, Reader in Mathematics in the University of Aberdeen.

For the third consecutive year, the number of entrants in the section was high. The total was 958 compared with 985 in 1997-98 and 1010 in 1996-97. The following table of the numbers of entries and entrants in 1998-99 reveals some of the interesting details. The figures in italics give the totals for 1997-98.

Number of schools taking part: Secondary 62 63

Primary 62 58

The reduction in the number of entries received in the Junior Division seems to be big enough to merit being called 'significant'. There was also a reduction in the Middle Division, which at first sight looks relatively smaller. However, the percentage reductions with respect to the numbers in 1997-98 (to one place of decimals) in the two divisions were

Junior Division ((432 - 352) × 100 ÷ 432 = 18.5%

Middle Division ((151 - 124) × 100 ÷ 151 = 17.9%

which suggests that the reduction in the Middle Division is relatively the same as in the Junior Division. And what about the figures for the Senior Division? In 1998-99 there were 88 entrants in the Senior Division of this section. Compare this with the figures for the seven previous years (working backwards from 1997-98) which were 15 20 21 18 28 18 26. Thus there were more entrants in the Senior Division in 1998-99 than there were in the four sessions from 1994-95 to 1997-98 inclusive. There were no evident reductions in standard, so that there were not only considerably more entries in the Senior Division, but also considerably more good entries. With encouragement from parents and teachers (already much in evidence) and perhaps some more tuition in how to tackle and to solve problems, standards could go yet higher. My message to S5 and S6 is this: congratulations on your efforts, keep responding positively to the challenge of the problems, always study the solutions which are sent out to schools in due course, so that you can learn as much as possible from the experience of taking part and, above all, never give in.

It has always been the case in Mathematical Challenge that a few entrants submit excellent work for Problems I, then disappear without trace. Why does this happen? Were the problems too easy? Did the problem sheets for later sets disappear? Reading the mark-sheets, it is distressing to see that there were some people who gained full marks in Problems I and then did not even submit one solution for Problems II.

Numbers of entrants who submit solutions for all the rounds in a competition like Mathematical Challenge can be illuminating. The following table gives (i) the total number of entrants E in a division or divisions, (ii) the number T who submitted entries for all three sets of problems and (iii) T as a percentage of E (i.e. 100T/E). The figures in italics are the corresponding figures for 1997-98.

Number of schools:

Probably most of the entrants who submit entries to all the sets of problems hope to achieve recognition in the form of a certificate - gold, silver or bronze. From the above table, there were 384 entrants in the Junior Division of Section I in 1998-99, of whom 148 submitted solutions to all three sets of problems. As the table below shows, 96 of the 148 qualified for an award: 22 gold, 32 silver and 42 bronze. Expressed as percentages, 38.5% of entrants submitted solutions for all three sets of problems, with 25% of all entrants qualifying for an award; 5.7% of the entrants being awarded gold, 8.3% silver and 10.9% bronze. It must be stressed that in assessing the awards we do not base the numbers of prize winners on such percentages.

Certificates awarded in the North Section, 1998-99

Of the 96 awards in the Junior Division, 40 were won by pupils in S1: 8 gold, 14 silver and 18 bronze. Of the 85 awards in the Primary Division, 9 were won by pupils in P6: 3 gold, 3 silver and 3 bronze. Two girls from Fraserburgh Academy attempted the problems in the Senior Division, even though one was in S3 and the other in S4. Both performed well; indeed the solutions presented by the girl in S3 were amongst the very best in the whole history of the competition. Mathematical Challenge does not aim to highlight young people of exceptional talent; nevertheless we are always delighted to see their work and we like to put on record the fact that the talent is there. Another significant award winner, from Robert Gordon's College, had, in his final year at the school, the satisfaction of qualifying for his sixth Mathematical Challenge mug; he was in the 'top class' for awards in each of his six years. Others have achieved this, but very few!

The award ceremony was held in the University of Aberdeen on Monday 14 June 1999. As in earlier years, Mathematical displays were on view in the Department of Mathematical Sciences, and again this proved to be very popular. The attendance exceeded our expectations, and those present included award winners from several places sufficiently far from Aberdeen to make the prospect of travel daunting, to say the least. Bayble School on the Isle of Lewis, the Sir E Scott School at Tarbert on the Isle of Harris, Kirkwall Grammar School, Plockton High School and Wick High School were some of the more distant places represented. Students' rooms in one of the Halls of Residence were used by some of those who had no option but to stay in Aberdeen for one or even two nights.

The Department of Mathematical Sciences was host for the day and provided lunch for 150. When heads were counted, there seemed to be more than this; we think that one or two hungry students, who possibly had been fasting over the examination period, may have wandered into the dining room by mistake. Everyone was welcome to attend Kenneth Brown's general talk on the attractions of Mathematics and the personal experiences of one who had become a Professor of Mathematics. He told us an interesting story of how a school teacher had misjudged the prospects of one of his pupils, which he assessed by estimating the length of the pupil's hair. We judged from the enthusiastic reception to the talk that the assessment had been inaccurate.

Professor Brown is a Vice-President of the London Mathematical Society, for several years one of the main sponsors of Mathematical Challenge. The North Section has its own supporters to help in local matters: a donation from BP Amoco helped to meet the expenses for travel and accommodation for people coming to Aberdeen to the award ceremony and having to cover the cost of long and expensive journeys. IBM UK Ltd., as in earlier years, financed the tea and biscuits that refreshed the crowd of about 200 between the talk and the presentation ceremony. Royal Mail continued to play an important part throughout the year in forwarding correspondence, including problems, solutions and mark-sheets, efficiently and courteously as well as post-free. We are greatly indebted to Royal Mail for this support, which, in financial terms, is equivalent to a handsome donation.

We owe a great deal to those who are prepared to spend long hours marking the entries. They have always been considerate in their awareness of the need to treat every entry seriously and evenly. When marking, credit must be given where it is due. A solution must not be treated as rubbish just because the method is much longer and more cumbersome than is necessary; if it is correct and works, then credit must be given. Thanks are also extended to teachers for their willingness to process Mathematical Challenge papers conscientiously. Much of what the National Committee for Mathematical Challenge does would not be possible without the assistance of teachers. Supportive parents are of course very important to us, but we do worry about the loss of talent that can arise when a promising pupil has to cope with a negative attitude, or even a hostile one, from home or from their peer group.

There are many others without whom Mathematical Challenge would not be able to operate. Those who suggest new problems, who have the interesting but time-consuming task of trying to do them and also to grade their suitability for the competition, are mostly very busy people, whose services are much in demand. It has to be remembered that a problem cannot be used unless we are sure about the solution. Ambiguities and errors in logic are not always obvious. We must ensure that there are no mistakes in our problem sheets or our solution sheets, since these are to be so widely distributed. The work is interesting but it can be stressful.

We thank all for their welcome support and encouragement. I hope that Mathematical Challenge, as a competition open to pupils in schools throughout Scotland and locally here in the North Section, will thrive for many years to come.

E M Patterson

Section 2: East and Central

The number of entries for the secondary school competition was broadly in line with last year. The expected growth in the Primary Competition has finally occurred with twice the number of schools and entries. The figures for 1998-99 (with those for 1997-98 in italics) were

Number of schools taking part: Secondary 58 60

Primary 58 29

In the secondary competition 407 pupils from 45 schools entered all three rounds; the corresponding figures for the Primary competition are 179 pupils from 39 schools.

Certificates awarded:

The prize-giving was held at an Open Day in the University of St Andrews on Tuesday 1st June. 114 pupils from 28 schools were invited and there was a good attendance of pupils, teachers , parents and members of the local committee. The guest lecture entitled 'A Mathematician's Holiday' was delivered by Dr John O'Connor of the University of St. Andrews. This highly illustrated presentation took us to the Alhambra in Spain to name but one tourist venue.

After lunch in New Hall, the prizes were presented by Pofessor Edmund Robertson of the University of St. Andrews. To finish the day, Dr O'Connor and other committee members led a Microlab session where one could design one's own personalised wallpaper!

I would like to thank Dr Brian Fugard on behalf of the committee for all his hard work over the past two years as chairman and wish him all the best with his future career in Kent.

F I P Smith

Section 3: Lothian and Borders

This year saw a significant change to this section with the Dumfries and Galloway schools being incorporated into the new Section V. As a consequence, the number of secondary and primary schools taking part was roughly three-quarters of the previous year's. This has meant the section has become a far more manageable proposition. One or two large schools failed to participate this year, which meant that the remaining numbers fell somewhat. Whether this was due to teething problems with the change or due to the number of markers these schools would be asked to provide is still unclear.

The numbers participating in the section this year were:

Problems	Junior	Middle	Senior	Secondary Total	Primary	Overall Total
I	353	124	23	500	243	743
II	255	77	20	352	242	594
III	218	80	13	311	239	550
Number of contestants	409	145	33	587	350	937
Completing all three sets:	154	50	10	214	148	362

Number of schools taking part: Secondary 45

Primary 32

A figure not often given is the percentage of schools that actually participate in the competition. For this region it is about 65% of the possible secondary schools and it would be nice to see this increase.

Our prize-giving was on Saturday June 19th in the Swann Building at the University of Edinburgh. Some 80 Gold and Silver award winning pupils together with 70 or so parents and teachers attended the ceremony with Dr Colin Aitken, Head of the Department of Mathematics and Statistics, presenting the awards. After refreshments a lecture "How to draw a Straight Line without a ruler" was given by Prof. Elmer Rees to the Middle and Senior pupils, together with parents and teachers. The Junior award winners participated in one of the two activities, "Bridges and Bouncing Balls", or "Piles of Tiles". The Primary sector were mailed their awards.

In each of the divisions the Gold cut-off was set at 87%, the Silver 73% and the Bronze 67%. For whatever reasons there were many high scores this year and the distribution of awards reflected this. Boroughmuir High School within Edinburgh distinguished itself by having the top award winners in each secondary division, quite an achievement.

The numbers of certificates awarded this year were:

I am indebted to my markers and colleagues who have aided me in this enterprise: Thank you.

H W Braden

Section 4: West

Since this was the first year of the new Section 4, there is no way of comparing this year's statistics with last year. The final figures for this section are shown in the table below. Bracketed figures, for interest, are those of 1994-95, the last year for which I was previously responsible.

Problems	Junior	Middle	Senior	Secondary Total
I	763 (839)	221 (304)	74 (45)	1058 (1188)
II	481 (470)	140 (115)	41 (24)	763 (609)
III	401 (334)	105 (102)	31 (20)	537 (456)
Number of contestants	888 (947)	235 (315)	74 (45)	1197 (1307)
Completing all three sets:	308 (249)	101 (81)	31 (18)	440 (348)

Number of schools taking part: Secondary 70 (107)

You will note that, apart from first round Juniors and Middles (and of course numbers of schools), the figures for those local authority areas left in Section IV (Glasgow, Argyll and Bute, North and South Lanarkshire, and East and West Dunbartonshire) are higher than for the whole of the then Strathclyde Region of just four years ago.

A total of 266 certificates were awarded: 62 gold, 68 silver and 136 bronze. Gold certificates were presented at a ceremony held in the University of Strathclyde on Tuesday 8th June. The successful entrants were welcomed by Professor Adam McBride, who then presented their prizes and entertained them with a discourse on some interesting properties of numbers. After a buffet lunch, Dr David Weir gave a talk on the four colour problem.

In conclusion I must thank the large number (over thirty) of teachers prepared to help with the marking. Without them, running this section would be an impossible task.

D G Weir

Section 4P: West Primary

This new division consists of all primary schools in Glasgow, Argyll and Bute, North and South Lanarkshire, and East and West Dunbartonshire. The Division has its own organiser, administration and prizegiving.

Communication was therefore made directly with each of the 650 primary schools via the usual authority distribution channels. Primary teachers were invited to help as markers. However, only two teachers volunteered. They were invited to form a committee with the organiser to run the competition. Both accepted!

Marking was taken over most expeditiously by enlisting the services of 72 BEd students who had elected to specialise in mathematics education. This was overseen and monitored by their tutors.

A total of 840 pupils from 85 schools submitted entries to the sets of problems.

19% of the entry received an award, distributed as shown:

All gold and silver award winners were invited to a prizegiving ceremony in mid-June at St.Andrew's Campus, Faculty of Education, University of Glasgow. A talk on 'Codes' was given by Gerry McKaig and, after a short workshop (with refreshments), certificates and mugs were presented to all those pupils attending by Professor B J McGettrick, Dean of the Faculty of Education.

G McKaig

Section 5: South West

Although both the Primary and Secondary competitions in this new section (covering Ayrshire, Renfrewshire, Inverclyde and Dumfries & Galloway) were organised by the Department of Mathematics and Statistics at the University of Paisley, most of the Primary entries were marked by staff and students in the Faculty of Education at the Craigie Campus. This eased the burden of finding markers and of distributing entries for marking, and we are grateful to the staff and students who participated. Hopefully, the students learned something from the experience!

The numbers participating in the section this year were:

Problems	Junior	Middle	Senior	Secondary Total	Primary	Overall Total
I	424	151	13	588	341	929
II	239	77	8	324	267	591
III	197	68	7	272	235	507
Number of contestants				615	400	1015

Number of schools taking part: Secondary 50

Primary 55

The numbers of certificates awarded this year were:

Since Section 5 covers such a large area, it was decided to hold award days at the University of Paisley's sites at Dumfries, Ayr and Paisley. Schools whose pupils gained awards were invited to bring along all interested pupils and their parents, and a total of over 200 attended the three events. Prizes were presented by the Principal of the University at Paisley, an Assistant Principal at Dumfries, and by the Dean of the Faculty of Education at Ayr. After the presentations, the Section Organiser, Elizabeth West, gave a talk entitled 'A First Encounter with Maps, Snowflakes and Crinkly Curves'. Soft drinks and a chance to chat rounded off a very pleasant afternoon on each occasion.

E West

An example of a novel and pleasing solution to problem S4 in Set III, 1998-99

The problem:

In the figure is the midpoint of the arc of a circle, is a point on the arc between and , and is the foot of the perpendicular from to . Show that is the midpoint of the path from to along the line segments and .

Solution from István Gyöngy (Boroughmuir H.S.):

Since is the midpoint of the arc, .

If we rotate the triangle around so that goes to , then line goes to line because the angles and , standing on the same arc, are equal. The right-angled triangles and are identical because they have the same hypoteneuse , and . Therefore BE=BE'. Thus .

In the figure is the midpoint of the arc of a circle, is a point on the arc between and , and is the foot of the perpendicular from to . Show that is the midpoint of the path from to along the line segments and .

Solution from István Gyöngy (Boroughmuir H.S.):

Since is the midpoint of the arc, .

If we rotate the triangle around so that goes to , then line goes to line because the angles and , standing on the same arc, are equal. The right-angled triangles and are identical because they have the same hypoteneuse , and . Therefore BE=BE'. Thus .