Thursday

December 18, 2014

December 18, 2014

Posted by **Cass** on Saturday, September 16, 2006 at 10:17am.

How many centimeters is the diamter?

Clue:The measure of the diameter is between 6 x 0.15 and 9 x 0.15.

Clue:The units digit and the hundreths digit are the same

Clue:If the diameter was 100 times longer, its measure would be a square number

Clue:The measure of the diameter is between 6 x 0.15 and 9 x 0.15.

Clue:The units digit and the hundreths digit are the same

Clue:If the diameter was 100 times longer, its measure would be a square number.<<

it is betwen .90 and 1.35. The next clue means it is either .90 or 1.31, or 1.21, or 1.11.

MULTIPLY 100 to each of those, and see if you get a perfect square.

I have a question, Matt and Maggie were building a staircase from wooden cubes. The first step consisted of one cube, the second steop consisted of two stacked cubes, the third step had three stacked cubes, etc. The cubes were each 1/2 inch tall and were a variety of colors. Maggie asked matt, if we had 50 , I wonder how many blocks we would need?

It's a good idea not to pigyback questions like this John. You should start a new question so we don't miss your post.

If I understand the question correctly, we want each stack of cubes to be a different color and we want 50 steps, or stacks of cubes, then how many total cubes would we need?

The number of cubes for each step follows the sequence

1,2,3,...

Thus we want the sum of 1 + 2 + 3 + ... + 50

There are a number of ways to determine this sums. This is the method Gauss used for this problem.

Write the sum forwards then backwards directly beneath it like this

1 + 2 + 3 + ... + 49 + 50 = sum

50+49 +48 + ... + 2 + 1 = sum

Then add the rows to get

51+51+51 + ... + 51 + 51 = 2*sum

On the left hand side we have 50*51, so

50*51=2*sum Now divide both sides by 2 and find the sum.

In general the sum of the first positive n integers is (n(n+1))/2. Here n=50

From clue 1 you can see the number is between .90 and 1.35.

From clue 2 you can see the sumber looks like 1.d1. We need to check if it could be 0.d0 too.

From clue 3 100*d=x^2 so 1d1 is a square or d0 is a square. If you run through the numbers 1 to 9 you'll see whether d0 can be a square or not, i.e. check 10,20,30...,90. Now go through the number 0 to 9 for 1d1 for d, i.e. check if 101, 111, 121, 131 are squares.

I only found one answer. Remember the units too.

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