Bill's Number Plates

Problem

Bill is making a small profit by designing and selling (through the appropriate channels of course) number plates. The only rule that he has to obey is that a maximum of six characters, numbers or letters, is permitted. (However, plus and minus signs, or any other mathematical symbols, are not characters that he can use.)

He decides to make up some number plates. For instance, he can make 10 by using ‘10’ or by ‘TEN’. In how many ways can he make up 12?

What number less than 20 can be represented in the most ways?

Teaching sequence

1. Introduce the problem by first getting the class to think about number plates.
   What is the number plate of the family car?
   What is the strangest number plate that you have seen?
   What is the funniest one?
2. State the problem.
   What easy number plates can you think of?
   How many ways can you think of making 1?
3. Let the students go into their groups to tackle the problem.
4. After a while get the groups to report back.
5. Allow students time to write up the results of their work
6. Send your answers to Derek.

Extension to the problem

Agree on abbreviations to promote work in a desired area. For example, if you have been working on bases other than 10, you might want to use B for ‘base’. If you have been thinking about powers, then use P for ‘to the power of’. Then see what numbers the class can get.

Solution

Clearly this is an open problem. We have no idea what your students might come up with but we will put all new answers here with the name of the students and the schools where they come from. So get in quick while it is still relatively easy to find new answers.

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We have received the following possible number plates for TWO. They were devised by Aaron Holden a year 8 student from Waiwera South. Note the clever use of OVER for a fraction; B4 for before; and BY for multiplication. Can anyone find some other ways to make up two or is this all there are?

TWO, 2, II, TO, TOO,
8OVER4, 6OVER3, 4OVER2, 2OVER1,
97B499, 96B498, 95B497, 94B496, 93B495, 92B494, 91B493, 90B492, 89B491,
88B490, 87B489, 86B488, 85B487, 84B486, 83B485, 82B484, 81B483, 80B482,
79B481, 78B480, 77B479, 76B478, 75B477, 74B476, 73B475, 72B474, 71B473,
70B472, 69B471, 68B470, 67B469, 66B468, 65B467, 64B466, 63B465, 62B464,
61B463, 60B462, 59B461, 58B460, 57B459, 56B458, 55B457, 54B456, 53B455,
52B454, 51B453, 50B452, 49B451, 48B450, 47B449, 46B448, 45B447, 44B446,
43B445, 42B444, 41B443, 40B442, 39B441, 38B440, 37B439, 36B438, 35B437,
34B436, 33B435, 32B434, 31B433, 30B432, 29B431, 28B430, 27B429, 26B428,
25B427, 24B426, 23B425, 22B424, 21B423, 20B422, 19B421, 18B420, 17B419,
16B418, 15B417, 14B416, 13B415, 12B414, 11B413, 10B412, 9B411, 8B410,
7B49, 6B48, 5B47, 4B46, 3B45, 2B44, 1B43, 0B42,
2BY1, 1BY2.

These suggestions for possible number plates for THREE were devised by Samantha Bucky, from Dunedin North Intermediate. She has used words for operations and also used some foreign words for three. Are there other ways to make three?

THREE, THR33, NO3, N03, THWEE, NUMBA3, NOIII
1PLUS2, 2PLUS1, 0PLUS3, 3PLUS0, 0PLUS3, 3PLUS0, 1AND2, 2AND1, 3AND0, 0AND3,
THR3E, THRE3,
3BY1, 1BY3, 3X1, 1X3,
9MNUS6, 8MNUS5, 6MNUS3, 7MNUS4, 5MNUS2,
III, IIANDI,
9TAKE6, 8TAKE5, 7TAKE4, 6TAKE3, 4TAKE1, TAKE0,
1ADD2, 2ADD1,
TORU, TROIS, DREI, TRE, TRES.

These suggestions for possible number plates for FIVE were devised by Adam Collins, from Dunedin North Intermediate. Are there other ways to make five?

5, 05, O5,
FIVE, F1VE, FIV3, F1V3, FIVE5, F1VE5, FIV35, F1V35,
HALF10, HALF1O, HLF10, HLF1O,
4PLUS1, 3PLUS2, 2PLUS3, 1PLUS4, 5PLUS0, 5PLUSO, 0PLUS5, OPLUS5,
4ADD1, 3ADD2, 2ADD3, 1ADD4, 5ADD0, 5ADDO, 0ADD5, OADD5,
SINCO, S1NCO, SINC0, S1NC0, RIMA, R1MA,
RIMA5, R1MA5, 5RIMA, 5R1MA,
NUMBA5, NUMBR5, NO5, N05,
V, IADDIV, IVANDI,
5BY1, 1BY5, 5TIMES, 5T1MES,
000005, 00005, 0005, 005, 05,
OOOOO5, OOOO5, OOO5, OO5, O5,
HI5, TAKE5, TAK35,
1B46, 2B47, 3B48 ... (and on to 94B499)

These suggestions for possible number plates for FOUR were devised by Room 1 at Greenmeadows Intermediate School Manurewa.

FOUR, 4, 04, 004, 0004, 00004, 000004, O4, OO4, OOO4, OOOO4, OOOOO4, FOURS, FOURZ, FOUR4, 4FOUR, 4FOUR4, 2PLUS2, 1PLUS3, 3PLUS1, 6B4TEN, 6B410, 5B49, 4B48, 3B47, 2B46, 1B45, 0B44, ONEB45, TWOB46, 2B4SIX, 7B411, 8B412, 9B413, 10B414, 11B415, 12B416, 13B417, 14B418, 15B419, 16B420, 17B421, 18B422, 19B423, 20B424, 21B425, 22B426, 23B427, 24B428, 25B429, 26B430, 27B431, 28B432, 29B433, 30B434, 31B435, 32B436, 33B437, 34B438, 35B439, 36B440, 37B441, 38B442, 39B443, 40B444, 41B445, 42B446, 43B447, 44B448, 45B449, 46B450, 47B451, 48B452, 49B453, 50B454, 51B455, 52B456, 53B457, 54B458, 55B459, 56B460, 57B461, 58B462, 59B463, 60B464, 61B465, 62B466, 63B467, 64B468, 65B469, 66B470, 67B471, 68B472, 69B473, 70B474, 71B475, 72B476, 73B477, 74B478, 75B479, 76B480, 77B481, 78B482, 79B483, 80B484, 81B485, 82B486, 83B487, 84B488, 85B489, 86B490, 87B491, 88B492, 89B493, 90B494, 91B495, 92B496, 93B497, 94B498, 95B499, 4OVER1, 4OVA1, 2TIME2, 4TIME1, Q4, QQ4, QQQ4, QQQQ4, QQQQQ4, 8MNUS4, 7MNUS3, 6MNUS2, 5MNUS1, 4MNUS0, 9MNUS5, 8MNU54, 7MNU53, 6MNU52, 5MNU51, 4MNU50, 9MNU55, 4TH, FOURTH, IV, 4AFTR0, 3AFTR1, 2AFTR2, 1AFTR3, 0AFTR4, 8DVD2, 12DVD3, 16DVD4, 4DVD1, 20DVD5, 24DVD6, 28DVD7, 32DVD8, 36DVD9, 2X2, 4X1, FOURSS, FOURZZ, 444444, 44, 444, 4444, 44444, FOR, FORS, FORZ, FORSS, FORZZ, FORSSS, FORZZZ, 3ADD1, 1ADD3, 2ADD2, FO_UR, F_OUR, F_OU_R, FO_U_R, F_O_UR, _FOUR_, _FOUR, FOUR_, F_O_R, _F_O_R, F_O_R_, FO_R, F_OR, _FOR, FOR_, _FOR_, F0UR, F0R, 4F0UR4, 4F0R4, 4F0UR4, 4F0R, F0R4, F0UR4, 4F0UR