Happy Numbers were introduced in
problem from June 1997.
I was first shown Happy Numbers twenty years ago. It works like
Chains of numbers are made. The first number in the chain is split
into individual digits, each digit is squared and the sum of these
squares becomes the next number in the sequence, and so the process
continues; splitting, squaring and adding.
For example, starting with 35 the next number is 34, that's 9 + 25,
the number after 34 is 25, followed by 29, 85 and so on.
Explore this process for yourself to see what happens.
Do numbers just get larger and larger, or is there a limit?
Do numbers eventually map back to themselves?
Using a spreadsheet for mapping
sequences of this kind
Sometimes a mathematical situation seems to have no particular
purpose but, if our curiosity is excited, and our minds become
energised, then exploring a situation's possibilities may reveal an
interesting structure or lead us to invent a technique that could
be useful elsewhere. That is what happened for me with this
investigation. I should admit that the Happy Numbers investigation
itself has never done much to arouse my curiosity. When I found
answers, my response was "OK, now I know. So what?" But the
investigation did lead me to try out an Excel function which looked
up results I had already calculated, and that was a worthwhile
Putting Excel to work
As always, the main value of the spreadsheet when investigating a
mathematical situation, is to give results quickly and easily, so
that as much of my attention as possible is available to think
about any patterns or connections I notice.
When I first explored Happy Numbers I didn't use a spreadsheet.
Actually twenty years ago spreadsheets were strange and new to most
people. In some ways that was a good thing. Using a computer can
frequently become the automatic choice. When we realise we have
become automatic in our response to a situation it can often be
helpful to pause and try out the options; at least in our
Twenty years ago, in my pre-spreadsheet days, I sat for a bit,
thought about the process I was being invited to explore, tried out
a few calculations, started recording some results, and so on. I
still usually start work this way.
After a little while I realised that I was spending too much time
calculating and not enough time thinking about what was happening
in the process. I did notice that the order of the digits was
unimportant, that starting at 35 or 53 was really the same thing
apart from the first number. I also invested ten minutes, away from
the main task, to made a table of results so that I could at least
stop doing all of the individual calculations and record my results
I produced the table back then by doing my own calculations, except
for the symmetric values of course. Now I would very likely use a
spreadsheet for a table of results from a function of two
to see how to do this.
The table of results also helped in another way. It helped me see
that all two-digit numbers mapped onto one- or two-digit numbers,
except those that mapped onto 100, 106, 113, 128, 130, 145, or 162,
but each of these immediately map back to a two-digit result, so
the set has closure.
It is also worth pausing to see what happens to numbers with plenty
of digits (they always reduce in size), and to understand why that
The three-digit number which produces the largest next value is
999, and this maps to 243, so all three-digit numbers map to
three-digit numbers until they fall into the two-digit set
The four-digit number which produces the largest next value is 9999
which maps to 324, so all four-digit numbers map to numbers with
less than four digits.
I said earlier that the Happy Numbers problem did make me think
about how to use a spreadsheet in a new way: to look up results I
had already calculated.
Here's how I did that:
Happy Numbers with VLOOKUP
Although the square table was useful when I had to fill in the
answers myself, because it saved me doing separate calculations for
both numbers in a pairs like 35 and 53, once a spreadsheet is doing
the calculations this saving is not of much value.
So I produced a table that had the numbers from 1 to 200 in the
first column, split these numbers into individual digits, which
were then squared and added up.
Here's the Excel file for the Happy Numbers
In the Excel file "Happy Numbers mapping" column A contains the
value to be mapped, columns B, C, and D contain the individual
digits, and column E contains the mapping.
The following explains how the digits are isolated.
You need to understand that the INT function takes the integer part
of a value and ignores the rest. INT of 3.845 is just 3.
B2 calculates the number of hundreds using the formula:
A2 was the original value, dividing it by 100 and taking just the
integer part of this result reports the number of hundreds in the
For example: 374 divided by 100 is 3.74 whose integer part is 3,
the hundreds digit.
C2 calculates the number of tens using the formula:
INT(A2/10) calculates the number of tens. For the example of 374
this would be 37.
INT(A2/100) calculates the number of hundreds, 3, as already
So 37 minus 10 lots of 3 gives the result 7, the tens digit as
D2 contains the formula: =A2-10*INT(A2/10)
INT(A2/10) is the number of tens, 37 in the example.
So the original number minus 10 lots of the number of tens, 374 -
370, gives 4, the required units digit.
Once I had these basic results available, Excel could look these up
as each sequence was created using starting numbers 1 to 200 .
Click here to see the Excel file Happy Numbers
Click here to see the same Excel file with
conditional formatting used to highlight the structure
Using Conditional Formatting makes values of interest easier to
to look at notes on conditional formatting.
When to use the LOOKUP
What made this problem different from some other sequence work was
that the mapping rule was not a single, simple formula to enter
into a spreadsheet cell.
Consider a very different example:
I can find one approximate solution to the equation $x^2 - 7x + 9 =
0$, capable of unlimited refinement, by using the sequence mapping
7 - ( 9 divided by the previous value)
It's easy to tell the spreadsheet to take a number, divide 9
by that number, and subtract that result from 7.
Happy Numbers was harder because the digits needed to be
Although this could have been part of the Excel formula, it
would have been a lot to bundle into a single cell, and the risk of
Done using the Lookup command instead, the process is easy to
understand, the results referred to are laid out in an
easy-to-follow fashion and the whole process remains
For this method it was also necessary that the results formed
a closed set. Each result leads to another result from that same
This kind of structure helps to keep a process clear, making
interesting relationships easier to spot, and explain.