30-maths-starters1

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30MathsStarters
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IntroductionThis is a collection of puzzles for Maths lessons. Hopefully you'll find these are a bit moreinteresting than the "Let's count up in 0.2' e" type of lesson starter that teachers in Englandare encouraged to use. It is also hoped that some of these puzzles can be used as a talkingpoint to introduce new mathematical ideas.There are plenty of old chestnuts here, along with some new puzzles. There are also a fewgeneric starters for when you're feeling lazy ...

Suggested uses: Print onto transparencies for lesson starters Print and laminate for extension or small group works Duplicate two to a page for homework

The star rating is a rough guide to difficulty:

~ Little teacher input is required. These puzzles don't require muchmathematics beyond basic arithmetic.~~ These may require explanations of key words by a teacher. They may require

more work than one star puzzles, or a particular insight to solve them.~~~ These puzzles can be difficult, and may need a lot of teacher input.

30Maths Starters page 3 www.subtangent.com/maths


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Find The Path

Start at the bottom leftsquare and move up, down,left or right unti I you reach

the finish.

894576 649 97 888 6

Add the numbers as you go.Canyou make exactly 53 ?



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Missing Number

Which number should go inthe empty triangle?3 6

2 414.....___......



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How Many Triangles? *3How many triangles can you

see in this picture?



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Times Table

There is something strangeabout this addition square.Can you work out what the

missing number is?

30Maths Starters

3 8 113 6 11

11page 7

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Missing MatchesRemove just 4matches to

leave 4 equilateral triangles -they must be all the same

size.

30Maths Starters pageS www.subtangent.com/maths


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The Jumping FrogA frog has fallen into a pit

that is 30m deep.

30m

Eachday the frog climbs 3m,but falls back 2m at night.How many days does it take

for him to escape?30Maths Starters page 9 www.subtangent.com/maths


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Number Pyramids ** 7Can you work out what

number wi II be at the top ofthe pyramid?

13 16 12

Can you make a pyramid with100 at the top?



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Number RaceHere are the first few square

numbers:I I I 4 I q I 16 I 25 I 36 I 4q I

Here are the first fewFibonacci numbers:I I I I I 2 I 3 I 5 I g I 13

The square numbers are wellin the lead. Do the Fibonaccinumbers ever catch up?

...and here are the triangularnumbers coming up on the

rails...I I I 3 I 6 10 15 I 21 I 2g I30 Maths Starters page 11 www.subtangent.com/maths


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Perfection

6 is a very special number.The factors of 6 are 1, 2, 3and 6.If we add the factors otherthan 6 we get 1+2+3=6.

Can you find another Rerfectnumber?

30 Maths Starters page 12 www.subtangent.com/maths


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Magic SquaresCanyou put the digits 1 to 9 ina square so that every row,

column and diagonal add to 151

This example doesn't work:1 3 59 6 42 7 8

~9

~19~17

~13 ~12 ~16 ~17 ~15



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GamesPool balls are numbered from 1 to 15.What is the total of the numbers onall the balls?

Dominoes have two parts, each canhave from 0 to 6 spots. How manydifferent dominoes are there in a

set?If you roll 2 dice and add the spotsare you more likely to get an even or

an odd number? What if you multiplyinstead?r : e l r - " I

l!:J~30 Maths Starters page 14 www.subtangent.com/maths


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Monty HallYou are the winner in a quizshow and can choose a prizefrom behind 3 locked doors.Behind one door is a new car.Behind the other two doors are

goats.

When you have made yourchoice the host opens one ofthe other doors to reveal a

goat.Should you stick with yourchoice, or switch to the otherone? Or does it make no

difference?30Maths Starters page 15 www.subtangent.com/maths


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Number Lines 1 **13

Canyou put the numbers 1 to 7in each circle so that the totalof every line is 121



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Number Lines 2Can you put the digits 1 to 11 inthe circles so that every line has

the same total?



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Words & NumbersThe number 4 in...

Norwegian,Dutch,

and Englishbut not in

French

firevier

four

quatrehave something in common.They all have the samenumber of letters as theirname (4).Can you find more numbers

like this?30Maths Starters page 18 www.subtangent.com/maths


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Four Twos Make Ten **16There are lots of ways of

making 8 with four 2s usingstandard mathematicalnotation. For example:

2+2+2+2, oreven 22+2;.2.Can you find another way?

Can you find a way of making9 with four 2s?

What is the biggest numberyou can make with four 2s?What is the smallest numberyou can't make with four 2s1



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Petrol Prices **17Some petrol stations displayprices by sticking segmentstogether to make numbers.

What is the largest numberyou can make with 10segments?

What is the largest you canmake with 16 segments?



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A Fishy TaleIf you write the word COD ina grid like this there are twoways of spelling the word.

~~~How many ways can youmake HAKE?H A KA K E

What about COLEY orSALMON?COL SAL MOLE A L M 0LEY L M 0 NCan you predict how many

ways there are to makeBARRACUDA?30 Maths Starters page 21 www.subtangent.com/maths


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How Many Squares? **19How many squares can you

see in this pattern?

How many rectangles arethere?



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The Konigsberg Bridges *20The city of Konigsberg hadseven bridges that crossed

the river Pregel.River bank (A)

R iver17ank (B )Can you find a way ofcrossing all the bridges

exactly once?You can't go over a bridge

more than once.



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Crooked Calculator ~ **21Eric has got all the sums wrong. Eachtime he pressed exactly one wrongkey.

II rn []] [I] rn []]G = : : ~ i l ; : : I I ; : : 1[]] [I] rn [I] rn G ::~::i. . ?[]] [I] []] [I] rn G ::~::

IIlD OJ[]] [I]OJG = : : i : i .Can you work out wh ich keys he

actually pressed?Eric manages to press the right keysbut gets them in the wrong order.Can you get the keys in the right

order?II rn rn [I] []][]]G i : : : = : i.[]][I]rnGrnG[]]G][]]@]rnG

IIlDrnOJ[I][]]G

. . . . . I1 : : : 1

. ? I



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Drinkers **22In the town of Ketterby

800/0 drink cola700/0 drink coffee500/0 drink tea

Is it certain there is someonewho drinks cola, coffee and

tea?



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Coin of the Realm *23These are the coins commonly

used in Britain today

What is the fewest number ofcoins you need to make

(a) 83p (b) 1.34 (c) 5.27?

What is the smallest amountthat needs more than 5 coins?30Maths Starters page 26 www.subtangent.com/maths


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Four Lines **24Put 9 dots in a square like this

Can you go through all 9 dotswith four straight lines?You can't take your pen offthe paper.You can start where you like.



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Back To Front

Write down a 2 digit numberI , 62 I ]

Reverse the digitsI i 2.6 I ]

Work out the difference62. - 2.6 = 36

Try more 2 digit numbers.Can you see a pattern?

What happens with 3 digitnumbers?30Maths Starters page 28 www.subtangent.com/maths


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St. IvesAs I was going to St. Ives,I met a man with 7 wives.

Eachof the wives had 7 sacks.

In each sackwere 7 cats.

Kits, cats, sacks,wives - howmany were going to St. Ives?



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FruitfulThere are three boxes.

One box contains pears, onecontains oranges, and onecontains pears and oranges.

Pears &Oranges

The labels have fallen off andall have been stuck back on

the wrong boxes.Barry opens one box andwithout looking in the boxtakes out one piece of fruit.He looks at the fruit andimmediately puts the labelson the the right boxes.How did he do it?



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Sum & Product A1Find 2 numbers whose

sum is and---product is .

30Maths Starters

+x+x

page 31 www.subtangent.com/maths


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Harry The Hedgehog A2



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Today's Number A3Today's number is

Add 17Double it

Multiply it by 10Halve itSubtract 7

Multiply by 6Square itFind its factorsFind 1Aof it



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Notes 6. The Jumping FrogIt will take him 28 days to escape. After 27days and nights the frog has only 3 metres to

1. Find The Path go. On the 28th day the frog is able to jumpHere is one solution: 4 9 7 7 4 clear.5+7+6+6+9+4+5

8 9 4 5 7 7. Number Pyramids+ 7 +4 = 53 6 6 4 9 9 57. Each number is obtained by adding theA tougher challenge is 7 8 8 8 6 two immediately below it. A good strategyfor solving these puzzles is to start with theto find a route to make 5 6 5 5 target number in place and to work60. downwards. This can clearly be extended to2. Missing Number non-integral and negative targets.The missing number is 5 (add the bottom 8. Number Racetwo numbers and divide by the top number). Th 12th d Fib . be square an 1 onacci num ers areStudents may be encouraged to come up b th 144 Aft thi th Fib . bo . er s e 1 onacci num erswith puzzles of their own.3. How Many Triangles?12 + 6 + 2 = 20

4. Times Table

+ 1 1 hours1 1 o 'c lo c k 1 0 o 'c lo c k

are in the lead. The 10th Fibonacci andtriangular numbers are both 55. After this thetriangular numbers are forever doomed to3rd place. A good opportunity to explore theFibonacci numbers.9. Perfection28 = 1 + 2 + 4 + 7 + 14. The next two are 496and 8128. The even perfect numbers arerelated to the Mer sen n e pr im es.2 P -1 is a prime if and only if 2 P -1 (2 P - 1) ISa perfect number. It is not known if there areany odd perfect numbers. You couldinvestigate multiple-perfect numbers, wherethe sum of the proper divisors of a number isan exact multiple of it.

11 + 11 = 10. This could clearly be extended 10. Magic SquaresHere is one solution:o other moduli. The more observantstudents may spot the clue in the title.

5. Missing Matches/~r.VVVHow may different solutions are there?

30Maths Starters

8 1 63 5 74 9 2

=15=15=15

=15 =15 =15Diagonals: 4 + 5 + 6 = 15, 8 + 5 + 2 =15.See Appendix A for a fairly simple method forconstructing magic squares of odd order.



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Continuing in the same way gives1 + 2 + 3 + ... + 7 = 28

_1 This is a little easier than the previousDice:2 puzzle. Put 6 in the centre. Then place 1P(odd on dice)=p(even)=~ opposite 11, 2 opposite 10, etc. to give totals

P (odd sum) =P (odd + even)+P (even + odd) of 18. Inboth puzzles the realisation that you1 1 1 1 1 1 1 should put the median number in the centre=-x-+-x-=-+-=-2 2 2 2 4 4 2 to maintain symmetry helps enormously.

Alternatively you could draw a sample space (You could also put 1 or 11 in the centre,diagram and get counting... since the remaining numbers can be paired

off in these cases to give totals of 14 and 22.

11. GamesPool: 120Pair off the balls 1+15, 2+14, etc.This gives seven pairs. 7 x 16 =112.Adding the 8 ball gives 120.Dominoes: 28List the dominoes with a blank (0):0-0, 0-1, 0-2, ..., 0-6 (7 dominoes)Then list those with a 1:1-1, 1-2, 1-3, ..., 1-6 (6 dominoes).We miss out the 1-0 as this is in the first list.

12. The Monty Hall Problem

13. Number Lines 1You could give the hint that the centre digitis 4.

14. Number Lines 2

You should switch (assuming you would 15. Words & Numbersprefer a car to a goat). It is tempting to An or th on ym ic n um be r is one that has theassume that the two remaining doors have same number of letters as its name suggests.an equal chance of hiding the star prize, but "Four" is probably unique inEnglish. (Howthis neglects the fact that the host's choice of would you prove it?) Some more exoticdoor to open is not independent of your examples include" queig" (Manx,S);initial choice. "bederatzi" (Basque, 9); "du" (Esperanto, 2)The probability of your initial choice being and" amashumi amabili nesikhombisa"correct is t .This probability doesn't change (Zulu, 27).when the host opens one of the other doors. An interesting question is to find expressionsSo the probability that the remaining door that are orthonymic, e.g. "five add seven" (12hides the prize is ~. You can find lots more letters).on the 'Monty Hall Problem' at 16. Four Twos Make Tenhttp://math.rice.edu/-ddonovan/montyurl.html

30Maths Starters

Two other ways of making 8 are 2x2 + 2x2and the less obvious (f+ 2)!+2 .222+2 =65536 may be the largest if we restrictourselves to powers and the 4 basicoperations. 7 is the smallest positive integerthat cannot be made without resource tofactorials etc. A calculator is useful.

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17. Petrol PricesIII II d III III II d fIIIIIan IIIIIIII - a goo way 0showing the importance of place value. Amore interesting question is to ask howmany dif ferent numbers can be made with 7segments.18. A Fishy Ta IeCOD: 2, HAKE: 3, COLEY: 6, SALMON: 10The formula for a word of length n is

( n - 1)! ,ifnis odd(n ~ 1 )! ( n ~ 1 )!

( n - 1)! , if nis evenn n("2-1)!("2)!So BARRACUDA gives 8!/(4!4!) = 70.This is discussed in Appendix B.19. How Many Squares?28 @i-t. 16 @2x2, 6 @ 3x3, 2 @4x4.Total: 5220. The Bridges Of KonigsbergIt can't be done. Look at this network:

D

21. Crooked Calculatorr e m . rrilri:lr-)('1 r0riSl r= .-.......... W t.fu~ W L . f u l . . = . . J :~I~JI~J m[B] ]OJ rn0 :~)?.[]]Q[]][Brn0 :~).1D[]][]][]][]]0 4l.rn[]][B[]][]]0 ei_[]][B[2JB[[]0 i a.[]]rnmQ[]]0.1D[]][]][B[]]0 " ?22. DrinkersNo - it is not certain. Suppose there were 10people in Ketterby. You could have:

T T T T T T T T T Tcola .I .I .I .I .I .I .I .Icoffee .I .I .I .I .I .I .Itea .I .I .I .I .I

20% is the maximum possible percentage ofpeople who drink none of the beverages.23. Coin of the Realm(a) 83p =SOp + 20p + lOp + 2p + lp(b) 1.34 =1 + 20p + lOp + 2p + 2p(c) S.27 =2 + 2 + 1 + 20p + Sp + 2p88p =SOp + 20p + lOp + Sp + 2p + lp needs sixCOIns.24. Four Lines

Every time we enter a piece of land we must You have to (literally) think outside the boxleave it by a different bridge. So there must to solve this old chestnut.be an even number of bridges attached to Here's one solution:each piece of land (except for the start andfinish). There are four pieces of land with anodd number of bridges. Even if we take thestart and finish into account there must beanother piece of land with an unused bridge.If you remove one of the bridges then itbecomes possible (such a path is called anE ule ria n pa th ). Does it matter which bridgeyou remove?You can find details of a visit by amathematician to Konigsberg athttp://www.amt.canberra.edu.au/koenigs.html

http://www.amt.canberra.edu.au/koenigs.htmlhttp://www.subtangent.com/mathshttp://www.subtangent.com/mathshttp://www.amt.canberra.edu.au/koenigs.html


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25. Back To FrontThe answers are all multiples of 9.It's not too difficult to prove this:The original number can be written asx= 10a+b where a and b are the twodigits. The reversed number is thereforey =10b+a and the difference isIx- yl=l( 10a+b )-( 10b+a )1=19a-9bl

=9Ia-bli.e. 9 times the difference in the digits of theoriginal number.For three digit numbers you will getmultiples of 99.26. St. IvesA trick question. Since you met the man onthe way to St. Ives, he and his entouragemust be coming from St. Ives. So the answeris just one - you. If you're feeling evil youmight want to conspicuously distributecalculators for this puzzle.If you do actually go through thecalculations you should get 7+72+i+74 =7+ 49 + 343 + 2401 =2800.This problem is very old indeed. It issupposed to date back to the ancientEgyptians:http://mathsforeurope.digibel.be/story.htm27. FruitfulBarry opened the box labelled "pears &oranges". This box must contain only pearsor only oranges. If he picks a pear then heknows the box he opens is "pears" and theother two boxes must be "oranges" and"pears and oranges". The one labelled"oranges" must be wrong so it is labelled as"pears and oranges". A similar argumentworks if he picks an orange.Insummary:He picks Pears Oranges P&OA pear Oranges P&O Pears

An orange P&O Pears Oranges

A 1. Sum and ProductI - I f tre are a ew sugges IOns:

sum product numbers10 9 1,915 50 5, 108 15 3,518 56 4, 148% 4 %,84 3% 1%,2%3 1.89 0.9,2.1

A little algebra shows that for sum sandproduct p the numbers are t (s~S2 - 4 p) .A2. Harry The HedgehogWrite a number in his belly ...A3. Today's NumberJust for fun try rt,
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