In this page, WTM will present some interesting activities similar to those that appear in another page (Digital Diversions). The difference, however, will be that here all ten digits — 0, 1, 2,…,9 — will be used. This is the reason the word “pandigital” is used in the title. (“pan” is a prefix that means “all, every one“.)
Our first discovery activity involves another popular “P” word: palindrome. We begin with a palindrome of 9 digits. Next, you need to double that number and carefully examine the result. If you know how to multiply by 2, do it that way. If not, just add the palindrome to itself, like so:
673454376 + 673454376 -------------
Either way is okay.
Now, look at that final result. Wow! Isn’t that strange?
Let’s try the same idea with these numbers.
1. 682454286 2. 763454367 3. 764353467 4. 862454268 5. 892151298 6. 971353179 7. 981252189 8. 982151289
Now, we have a unique situation this time. The three palindromes listed here are the only ones that behave in our pandigital way upon performing the doubling process twice. Yes, double each number, then double that result again.
1. 481262184 2. 672393276 3. 673454376
(Note: If you don’t like to double, then double again, what can you do instead?)
It is rather appropriate that for this 3rd activity in the series that our little factor should be 3 itself. So try multiplying the palindromes below by 3 to see what happens. Be careful.
1. 345282543 2. 345828543 3. 543282345 (note the digits) 4. 567828765 (note the digits) 5. 782353287 6. 984393489 7. 935131539 8.1357227531 9.1567227651
Are you ready for a new twist to our marvelous multiplying math idea? This time we’ll use two palindromes – a big one, of course, like before, but also a small one. In fact, our small one is the smallest, non-trivial palindrome of all: 11.
So, what are the pandigital products of these beautiful numbers multiplied by 11?
1. 189414981 2. 369252963 3. 567252765 4. 589010985 5. 765414567 (note the digits)
Something interesting occurs if we change our rules of our basic game a little bit. Observe the following:
2765115672 is a nice palindrome of ten-digit size. (Remember: that’s not ten different digits.) If we multiply it by 3, we get 8295347016, which is a nice pandigital number. All ten digits are present.
But what if we had written down our palindrome hastily, omitting one of the pair of 1’s that are in the middle? It could happen, you know. We do make mistakes from time to time, whether we’d to admit it or not. What affect would this produce on our product? Let’s see.
276515672 x 3 = 829547016
Ta-dah! An interesting value, don’t you think? We have a 9-digit number composed of 9 different digits! And look closer – who’s missing? Who is absent? Why, the digit 3, of course. Now that’s truly interesting.
But is it a once-in-a-lifetime event? Well, lucky for us numerophiles, it isn’t. In fact, there’s another one nearby. This time the palindrome to be shortened by removing a central digit of 1 is 2753113572.
First, try multiplying it by 3 to see the true pandigital product. Then multiply the shorter version by 3 again. Who’s absent this time? (Hint: it’s not 3.)
And our luck continues if other numbers besides 3 are used as our factor with a reduced size palindrome. Here’s a small listing of some dual-use palindromes with their accompanying factor:
1. 12 x 87399378 2. 12 x 147111741 3. 12 x 251383152 4. 13 x 82533528 5. 15 x 95311359 6. 15 x 275131572 7. 17 x 92977929 8. 27 x 172585271 (*)
Remember: always remove the middle digit, or one of a middle pair of identical digits.
(* Special note: the palindrome here is not just any old palindrome. It also happens to be a prime number. Hence, it’s called a palprime, for short.)
Some Division Trivia
The number statement given in red in the previous activity provides us with a new angle to pursue. Using the idea of inverse operations, we can change the multiplication into a division like so:
276515672 x 3 = 829547016 829547016 ÷ 3 = 276515672
We might notice that the left side, the division, now consists of all ten digits, right? Hence, it is a pandigital division statement. This nice idea should prompt any good number hunter to look for more such cases.
We’re happy to report that many do exist, and they come in different categories according to the number of digits in the dividend and the divisor. With the help of our collaborator, Jean-Claude Rosa, leaving the palindromic quotients as an exercise for you, we present the following summary:
A. abcdefghi ÷ j the smallest: 128943760 ÷ 5 the largest: 862504731 ÷ 9 number of solutions:146 B. abcdefgh ÷/ ij the smallest: 12905376 ÷ 48 the largest :98346501 ÷ 27 number of solutions:140 C. abcdefg ÷ hij the smallest: 1293864 ÷ 507 the largest: 9857016 ÷ 234 number of solutions: 95 D. abcdef ÷ ghij the smallest: 105468 ÷ 2397 the largest: 972630 ÷ 4815 number of solutions:118 E. abcde ÷ fghij the smallest: 26970 ÷ 13485 the largest: 98760 ÷ 12345 number of solutions: 94
Pandigitals and Pi
Every school boy and girl, at least in my school days long ago, learned about the famous number “pi”. Of course, the value of this fascinating number is approximately 3.14, but we were often taught about a fractional approximation of 22/7, or the mixed number version 3 1/7. This leads us to wonder if we could connect our topic of pandigitality with pi. Here is what we have come up with so far.
First, let’s examine 22/7, the so-called improper fraction form. Using all ten digits, we can show these three representations:
49302 56034 62370 -------- -------- -------- 15687 17829 19845
(In fact, these are the only ways possible with all ten digits.)
Turning now to the mixed number form, we are faced with a slightly different situation. The whole number part, 3, is fixed and can’t be changed, whereas the fraction, 1/7, can take many forms.
This means we need to find ways to write 3 n/d, where n & d are composed of the remaining nine digits. Unfortunately, such is not possible; there do not exist two numbers of the form abcd and efghi, such that
abcd 1 3 ------- = 3 --- efghi 7
Hence, we must be content with cases that use fewer digits. The only one that uses 8 digits in the fraction is
1094 3 ------- 7658
The smallest case is 3 2/14, which is interesting in its own right as it is made from the first four counting numbers (a.k.a. the natural numbers). There are five other cases where the numerator is a single digit. That means, fractions of the form a/bc.
As we increase the number of distinct digits, we can state these following observations:
- There are 15 cases where the numerator is a 2-digit number. We leave it as an exercise to you, dear reader, to find those fractions.
- There are 3 cases where the fraction takes the form abc/def.the smallest is 104 / 728the largest is 108 / 756Again we will let you have the pleasure of finding the middle one.
- There are 15 cases where the fraction takes the form of abc/defg.the smallest is 208 / 1456the largest is 972 / 6804Since the region to search now is considerably larger – 200 to 999 – we won’t ask you to find them, unless you really want to.
Well, what do you think about pi now?
The numerical data used in some of the activities above come from Jean-Claude Rosa, a mathematician and school math teacher from France. A summary of his work appears as WONplate 114 in the World!OfNumbers.
Others come from the work of Patrick De Geest and likewise appear in his World!OfNumbers