NCETM_Mathematics_Department_Workshops_Index_Numbers_Resource_Sheet_HT2.IDX.4

Mathematics Department Workshops Topic: Index Numbers Resource Sheet HT2.IDX.4 Friedman Numbers Prelude: The Two Fours Puzzle Invite the learners to make as many different answers as they can using just two fours, each raised to any power, with one arithmetical operation in between. The challenge is to make the whole list from 1 to 10. There are many solutions, of which this is one: 1 = 40 × 40 1 2 = 42 × 40 1 3 = 42 + 40 4 = 41 × 40 5 = 41 + 40 1 6 = 41 + 4 2 3 7 = 42 − 40 1 8 = 41 × 4 2 3 9 = 42 + 40 31 10 = 4 2 + 4 2 Friedman Numbers ‘Friedman numbers’ are positive integers which can be made from the number’s own digits using one or more of • The four operations +, −, ×, ÷ • Exponentiation (powers) and brackets For example 126 = 21 × 6 and 625 = 5(6−2) so 126 and 625 are ‘Friedman numbers’. All the number’s digits must be used. These numbers are called ‘Friedman numbers’ after Erich Friedman, an American mathematician who discovered them. Find out about Erich Friedman from his homepage at http://www.stetson.edu/~efriedma/ Main Activity Introduce the idea of Friedman Numbers to the learners using the given two examples. Invite them to look for others. If necessary, ask them to consider whether or not 25 could be a Friedman Number. www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

When they comment that it is probably quite time consuming to look for them, invite them to consider the list of Friedman Numbers found under 10 000: 25 1260 2506 4106 121 1285 2507 4167 125 1296 2508 4536 126 1395 2509 4624 127 1435 2592 4628 128 1503 2737 5120 153 1530 2916 5776 216 1792 3125 5832 289 1827 3159 6144 343 2048 3281 6145 347 2187 3375 6455 625 2349 3378 6880 688 2500 3685 7928 736 2501 3784 8092 1022 2502 3864 8192 1024 2503 3972 9025 1206 2504 4088 9216 1255 2505 4096 9261 Learners can now be invited to explain why each of (the first 12 or so) Friedman Numbers qualifies for the title. Going Further Enter this into your calculator: (86 + 2 × 7)5 − 91 34 What Friedman number do you get? More information can be found at http://www.stetson.edu/~efriedma/fn/index.html The Friedman numbers are sequence A036057 found at http://www.research.att.com/cgibin/access.cgi/as/nnjas/sequences/eisA.cgi?Anum=036057 of Sloane’s ‘Encyclopedia of Integer Sequences’, at http://www.research.att.com/~njas/sequences/index.html www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching