Musings on mathematics and teaching.

## Month: July, 2012

### Arranging numbers in a grid

The National Council of Teachers of Mathematics is the largest organization for math educators in the United States, and they post a math problem on Twitter every Friday. Usually, these problems are not very difficult, but the problem for February 17, 2012 was quite challenging.

The definition of “touching” was later clarified for the purpose of this problem. Two squares are considered to be touching if they meet along an edge, or if they just meet at a corner point. For example, the first solution in this picture is valid, but the second is invalid, because the consecutive numbers 4 and 5 are touching at a corner.

I challenge my readers (all three of you) to find the number of solutions. You will probably need to write a computer program to do this. I will post my solution on July 31.

### Natural number impostors

The obvious answer is 10, but Dave’s answer was 11 because he had in mind the sequence of palindromes in base 10.

There are infinitely many ways to continue any given sequence, so the question has infinitely many “correct” answers. I looked up the sequence 1,2,3,4,5,6,7,8,9 in the On-Line Encyclopedia of Integer Sequences (OEIS), which is an enormous database containing over 200,000 integer sequences. This search returned 1825 sequences, among which were A000027 (the natural numbers) and A002113 (palindromes in base 10).

I wondered how many of these sequences continued to 10, so I looked up the sequence
1,2,3,4,5,6,7,8,9,10. This search returned only 1088 sequences. Based on this result, I predicted that there was a 60% chance that the next number in Gale’s sequence was 10. (1088/1825 = 0.596.) Alas, my guess was incorrect.

This raises the following question. How many of the sequences in the OEIS are “natural number impostors”? That is, if a sequence contains the first n positive integers consecutively, then what is the probability that the next term is n+1? Having formulated this question, which has no importance whatsoever, I was nevertheless driven to find an answer. The results are shown below.

The abrupt decrease from = 9 to = 10 is probably explained by the fact that many of the sequences in the OEIS are defined in terms of their base-10 representations. My raw data is listed below.

```n	a(n)
1	133108
2	56891
3	16940
4	6931
5	4239
6	3084
7	2467
8	2090
9	1825
10	1088
11	761
12	615
13	498
14	437
15	386
16	320
17	286
18	264
19	249
20	230
21	207
22	198
23	188
24	180
25	169
26	164
27	159
28	154
29	140
30	132
31	125
32	117
33	110
34	105
35	103
36	100
37	93
38	91
39	89
40	85```

### Squared rectangle

The picture shows a rectangle that has been subdivided into nine squares. The area of the smallest square is 1. Can you find the areas of the other squares?

Addendum: Anna Blinstein informs me that she has used this problem with great success as an end-of-year project for Algebra 1. Squared rectangles also make an appearance in the Exeter math problem sets.

### Whole numbers and place value

I am creating slide presentations (using PowerPoint) for the developmental mathematics class that I will teach in Fall 2012. The first presentation is a review of whole numbers and place value. One of my goals was to help students to visualize large numbers. Comments are welcome.