Number plates

The Problem

Billie is designing, making and selling number plates. Each can have a maximum of six characters, numbers or letters. Billie cannot use plus and minus signs, or any other mathematical symbols. He can make 10 by using ‘10’ or by ‘TEN’.

In how many ways can he make 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 your 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?
   How could you systematically figure out new number plates? (You might show some examples from the solution).
3. Have students work on the problem, keeping track of their ideas.
4. As the groups to report back, have them add their ideas to a growing class list. 

Extension

Work in a base other than 10, using B for ‘base’. 

Solution

This is an open problem for which there are very many solutions.

For example:

Number plates for TWO  (devised by Aaron Holden a year 8 student from Waiwera South). Are there any other ways to make two?

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.

Number plates for THREE (devised by Samantha Bucky, from Dunedin North Intermediate). Are there any 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.

Number plates for FOUR (devised by Room 1 at Greenmeadows Intermediate School Manurewa.) Are there any other ways to make four?

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

Number plates for FIVE (devised by Adam Collins, from Dunedin North Intermediate.) Are there any 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)

Number plates for SIX (devised by Room 26 at Ellerslie School.) Are there any other ways to make six?

6, 06, 006, 0006, 00006, 000006, 
O6, OO6, OOO6, OOOO6, OOOOO6,
VI, SIX, 6OVER1, 
3PLUS3, 2PLUS4, 5PLUS1, 4PLUS2, 1PLUS5, 6PLUS0, 0PLUS6, 
3BY2, 2BY3, 6BY1, 1BY6, HALF12, 
9TAKE3, 8TAKE2, 8SUB2, 14SUB8, 7SUB1, 13SUB7,
7TAKE1, 10SUB4, 11SUB5, 12SUB6, 15SUB9, 5AND1, 1B47, 1AND5, 6AND0, ONEB47, 
6X1, 1X6, 2X3, 3X2, 
SECHS, ONO, ZES, SES, ALTI, SEKS, ANIM, ENAM, SEIS, LIX, SEI, CHA, CHWECH, SITA, SASE, KUUS, KUUSI, ISII, 
1ADD5, 5ADD1, 2ADD4, 4ADD2, 3ADD3, 6ADD0, 0ADD6, OADD6, 6ADD0
1B47, 2B48, 3B49 ... (and on to 93B499)