Here ’s a classic brainteaser that I do n’t like . What ’s the next telephone number in this sequence : 1 , 11 , 21 , 1211 , 111221 , … ? The answer is 312211 , because each phone number describes the finger in the number that precedes it . We unfold with 1 , an arbitrary pick , but the next number describes 1 as “ a undivided one , ” i.e. “ one one , ” i.e. 11 . The next entry describes 11 as “ two single , ” or 21 . This , in good turn , is “ one two followed by one one , ” or 1211 , and so on .
Legendary mathematician John Conway studied this so - scream “ count - and - say ” sequence and really proved some interesting termination about it . It intelligibly die on incessantly , and the numbers grow to infinity , but surprisingly no digit other than 1 , 2 , and 3 ever appear . If you keep key out bigger and bigger numbers in this means , you ’ll never generate a bowed stringed instrument of four ones ( or 2 or threes ) in a quarrel . Conway also hit the books the chronological succession that leap from different start telephone number other than 1 . He proved that no matter what whole number you open with , the resulting successiveness will diverge to infinity … except for one . Determining which one is your incentive puzzle this calendar week .
I like the idea of number describing other numbers , but I ’d prefer it not range as a puzzle to solve . My gripe with sequence puzzler is that they ’re undetermined to multiple potential solutions . You could sure as shooting cook up some unusual mathematical operation that produces the same first five numbers as the look - and - say sequence but then deviates from there . Your main puzzle this week concerns a figure that key itself . And quietus secure it has only one result .
Did you miss last week ’s puzzle ? Check it outhere , and find its solution at the bottom of today ’s article . Be careful not to read too far forrader if you have n’t solved last week ’s yet !
Puzzle #39: A Self-Referential Number
Only one 10 - finger identification number has the following property . Its left over - most digit is the turn of 0s in the act , the next digit is the number of 1s in the telephone number , the next is the issue of 2s , and so on until the decently - most digit , which is the number of 9s in the number . Find the numeral . number ca n’t start with a zero .
An example of a four - fingerbreadth act with this property is 2020 . The first digit indicate that the number contains two 0s , the next indicates zero 1s , the next indicates two 2s , and the last suggest zero 3s .
fillip : you’re able to seed the search - and - say sequence with any whole number . For example , if you started with 39 , then the next launching would be 1319 ( one three , one nine ) . Conway demonstrate that all seeds yield a chronological succession whose ingress grow to eternity , with only one exception . Find the exception .
I ’ll be back next Monday with the solutions and a new puzzle . Do you acknowledge a cool puzzle that you think should be featured here ? Message me on X@JackPMurtaghor email me at[email protected ]
Solution to Puzzle #38: Tax Evasion
Shout - out to 8×10 for a swift answer tolast week’stax dodging teaser . I hope the IRS does n’t monitor these …
you may win a maximum of $ 50 in The Taxman Game . See the turns below :
You take $ 11 and the Tax Collector takes $ 1 ( 1 is the only useable element of 11 )
You take $ 10 and the Tax Collector takes $ 2 and $ 5
You take $ 9 and the Tax Collector take $ 3
You take $ 8 and the Tax Collector takes $ 4 ( $ 2 was already taken on move 2 )
You take $ 12 and the Tax Collector takes $ 6
You ’re out of legal move so the Tax Collector get hold of the concluding assay of $ 7
Your profits total $ 8 + $ 9 + $ 10 + $ 11 + $ 12 = $ 50 .
Now the very end of the game will postulate the Tax Collector taking the $ 7 payroll check no matter what happens , because you’re able to never take it for yourself and no multiples of 7 are available to make the Tax Collector take it in the beginning . So effectively three paychecks are out of play ( $ 11 , $ 1 , and $ 7 ) , leaving nine remaining . you could not get more than four of these nine because the Tax Collector must get pay on every good turn . The scheme we establish gets you $ 12 , $ 10 , $ 9 , and $ 8 , the four large left over paychecks . So our approach can not be improved .
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