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Happiness Is Integral Hapies Is Inegral But Not Rational Austin Schmid/unsplash.com Andre Bland, Zoe Cramer, Philip de Castro, Desiree Domini, Tom Edgar, Devon Johnson, Steven Klee, Joseph Koblitz, and Ranjani Sundaresan classic problem in recreational mathemat- 1970, pages 83–84) for a proof of this fact. ics involves adding the squares of the digits A number is called happy if it eventually reaches 1 of a given number to obtain a new number. and unhappy otherwise. For example, from the number 49, we obtain In this article we explore happiness in several dif- ferent numeral systems. We include exercises along AAlthough this action may seem a bit arbitrary, some- the way that we hope will enhance your experience. thing interesting happens when we do it repeatedly. The The proofs-based exercises are meant to extend the sum of the squares of the digits in 97 is ideas we present and are completely optional. We also Continuing in this way, we obtain the sequence, include a few example-driven exercises that we strongly encourage you to complete, as they will be important At this point, the sequence produces 1s forever. later in the article. The optimistic mathematician might hope that this Everyone Can Find Happiness will always happen, but alas, it doesn’t. For example, if we begin with 2, we obtain As we have seen, only some numbers are happy. Maybe this seems unfair. However, the definition of happiness depends upon the method we use to represent posi- Once we reach 4, we enter a cycle that returns to 4 tive integers: We rely heavily on the base-10 notation every eight steps. of our number system. Perhaps working in a different Amazingly, these are the only possibilities: The base would allow different numbers to be happy. sequence starting with any number leads to 1 or to the Recall that if is an integer, then any nonnega- cycle tive integer can be written uniquely as a sum That is, every natural number has a path that reaches one of the components in figure 1. See Ross where each digit ni belongs to We will Honsberger’s Ingenuity in Mathematics (Random House, express the base-b representation of a number as a list 8 September 2017 : : Math Horizons : : www.maa.org/mathhorizons mh-08-11-edgar-final-cx.indd 1 9/29/17 12:52 PM 129 28 61 97 106 82 86 68 73 40 79 37 680 130 307 100 32 58 16 20 101 1 10 13 85 89 4 2 Figure 1. Every natural number has a 23 1,001 1,000 31 145 20 path that leads to one 42 of these components 13,666,669,999,999,999 1,125 98 24 under repeated summing of digital 154 squares. with its least signifi cant digit written fi rst: is less well known that we can represent integers using fractional bases. Specifi cally, if p and q are relatively prime integers For example, in base 10, and then any nonnegative integer has a unique Hap i e s Is In egral representation of the form and in base 2, where each digit ni belongs to In partic- But Not Rational ular, when we recover ordinary base-p represen- Consequently, we can explore the digital sum of tations. Again, we will write the base- representation squares for numbers represented in diff erent bases. of a number as a list Let’s introduce the base-b digital sum of squares This seems as if it shouldn’t work. But it does! As function, Sb(n). So, for instance, our original base- an example, 10 function yields and in base 2, Notice that regardless of the base b, the sum of the squares of the For the rest of this article, we will focus on base- digits is computed in base 10. representations of numbers. Once again, a number is base-b happy if it eventu- Exercise: ally reaches 1 upon repeated application of the base-b Take a break, and get your hands dirty! digital sum of squares function. Generate the base- representations of the integers Let’s try this out on 20 using the base-2 digital sum from 1 to 14. Check your answers in table 1 (where the of squares function: subscript has been omitted for appearance). 3 -representation 3 -representation n 2 n 2 So, 20 is base-2 happy, even though it is not base-10 1 [1] 8 [2,1,2] happy (it is in the cycle containing 4). In fact, every positive integer is base-2 happy. Similarly, every num- 2 [2] 9 [0,0,1,2] ber is also base-4 happy. 3 [0,2] 10 [1,0,1,2] Exercise: Prove that every number is base-2 happy 4 [1,2] 11 [2,0,1,2] and base-4 happy. 5 [2,2] 12 [0,2,1,2] Exercise: How many base-3 unhappy numbers can you fi nd? 6 [0,1,2] 13 [1,2,1,2] A Rational Extension 7 [1,1,2] 14 [2,2,1,2] Although it is well known that we can represent num- bers in base b when b is an integer greater than 1, it Table 1. Base- representations of 1 to 14. www.maa.org/mathhorizons : : Math Horizons : : September 2017 9 mh-08-11-edgar-final-cx.indd 2 9/29/17 12:52 PM By inspecting table 1, we can glean a few interesting 2 24 properties of base- representations of positive inte- 23 16 3 gers, all of which are true in general. First, in contrast 4 21 17 7 to ordinary base-b representations when b is an integer, 22 14 13 10 6 not every digit sequence yields an integer in base- 5 20 18 15 For example, More specifi cally, if 8 25 then and is the remainder when we divide n by 3. Moreover, 9 26 11 is an integer, and its base- representation 27 12 is 18 For example, 20 = [2, 0,2,1,2]3/2. As promised, the Figure 2. All numbers except 1 eventually base- representatio n of 20 ends with a 1 and a 2, and enter the cycle under the when we divide 20 by 3 we obtain a remainder of 2. repeated sum of squares in base- Moreover, number eventually enters the cycle, then, because eventually enters the cycle, and hence so does n. Exercise: Prove that these properties hold for base- Let’s prove the claim that for all representations of integers greater than 5. Let First, Is Happiness Rational? notice that Now we now give a proof by induction that Now we can play the happiness game as we did in You already checked this by hand for the base-b by repeatedly applying the base- digital sum base cases So, we may assume of squares function, which we denote and that for all For example, starting with 10 gives the sequence Remember that which, for ease of notation, we denote as m. So, which then enters the cycle Let’s try a diff erent example: Since and we see that However, m is an integer, so Also, since Because we’ve reached 10, the previous example tells us that this sequence will eventually enter the cycle Therefore, By our inductive assumption, as well. we know that Thus, continuing our earlier calculation, Exercise: Gather more data. What happens to each number in table 1 when we repeatedly apply The fi nal inequality comes from the fact that In your experimentation, you should have found that every number other than 1 eventually enters the cycle Note: If you couldn’t complete all the earlier exer- (see fi gure 2 for a visualization). Is cises on your fi rst pass, can you steal inspiration from 1 the only base- happy number? Does every num- the techniques presented in this section to complete ber other than 1 suff er the Sisyphean fate of cycling those proofs? for eternity? In doing these exercises, you may have also noticed Further Questions that for n-values 9 through 14, If this It seems that base- is not a particularly happy base, inequality holds for all then we would know since 1 is the only base- happy number. Regardless, that every number other than 1 eventually enters the it is still surprising that all other numbers eventually cycle Indeed, if we assume that every converge to a single cycle. 10 September 2017 : : Math Horizons : : www.maa.org/mathhorizons mh-08-11-edgar-final-cx.indd 3 9/29/17 12:52 PM We conclude with a few open questions about happy of times must be applied until we enter the cycle numbers that might lead to great student research Are there numbers with arbitrarily 3 ? projects. large height in base-2 1) Two extensions of the digital sum of squares func- We have seen that being both rational and happy is tion are the digital sum function and the digital sum of a rare phenomenon in this context. Nonetheless, the cubes function, in which we take the sum of the digits properties of the digital sum of squares function in or cubes of the digits of a number, respectively. More nonintegral bases seem worthy of further study. We generally, Helen Grundman and Elizabeth Teeple ex- hope our questions have inspired you to play around tended this to studying the sum of dth powers of digits with rational base representations and ask your own for any (Generalized happy numbers, F ibonacci questions as well.
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