A Simple Mat(h) Problem

Last updated on May 22nd, 2019 at 06:44 pm

Interesting math problems have always seemed to jump out of the woodwork at me. Here’s a simple geometry problem inspired by mat cutting. You don’t need a mat cutter or mat cutting experience to solve this problem, however.

Starting with a regular 40″ by 60″ foam board, a 23″ (by 40″) slice had already been removed for another project (shown as the large black-hashed area on the left edge of the illustration). In the blue dashed lines of the illustration, I drew simple plans to cut out four standard 16″ by 20″ pieces, but then discovered that there were problems along the top edge requiring me to remove a one-inch strip (shown with red hash marks), leaving a piece of foam board 39″ high by 37″ wide.

Illustration for Mat(h) problemThe question is “How many 16″ by 20″ pieces can I still get out of this remaining foam board?”. One would make the cuts on their mat cutter with a razor-like blade, so you don’t have to worry about a kerf (the extra material removed by the width of the saw blade).

The seven best answers will receive $7 off any print and another $7 off if you choose to frame (or gallery-wrap) the image. I will publish some responses, but obviously not immediately. So that nobody dies from the suspense, we will put a two-month deadline on this offer. Prizes may be redeemed any time afterwards.  Good luck!

Another Method For Adjusting A Logan Sander

Last updated on September 15th, 2019 at 05:02 pm

To download a printable version of this article (AdjustingSander.pdf), click here.

We have a Logan Precision Sander Elite Model F200-2 disk sander for improving saw-cut miters for your picture frames to a “perfect 45°” after cutting the moulding to size on our miter saw. To maintain such perfection requires due diligence and occasional adjustment.

How Do You Know When It’s Time To Adjust Your Sander?

  1. You may notice that when you put your frames together, there is a small gap between the pieces of moulding either on the inside of all four corners or the outside of all four corners. If some corners have gaps on the inside and some have a gap on the outside, you have other problems.
picture frames showing that your sander needs adjusting
The sander used on the miters of these two frames needs to be adjusted. Note gaps in corners.

In the figures used in this article, the symptoms have been exaggerated for illustration purposes. If the condition of your sander gets this bad without you noticing, you may want to consider another profession or hobby.

  1. When you are comparing the lengths of opposite pieces, and you have them side by side with the miters face up and their back sides touching, you may notice by running your finger over the miter that they are the same length on one end of the miter but not the other, or that one piece of moulding is higher at one end of the miter and the other piece is higher at the other end.
Differences in the miter cuts of moulding
The differences (gaps) in the mitered ends of these pieces of moulding show that your sander needs adjusting

In either of these cases, it’s time to adjust your sander.

But What About The Miter Saw?

It may be true that the miter saw also needs adjustment, but that would have minimal impact on your frames because even if the angle of the cut was wrong, the sander should correct that problem. Of course, it would take more sanding to correct, which besides taking more time and effort could, in the worst case, result in your frame being too small, so it should periodically be checked and corrected according to the manufacturer’s instructions (I currently have no improvements or suggestions for that process). An indication that the miter saw needed adjustment would be if as you are sanding the miter, sawdust builds up on top of one side of the moulding faster than it accumulates on the other. If it takes too many revolutions of the sander to perfect the edge, that could also be a clue, or it could be time to change the sandpaper.

How To Adjust The Sander

On the last page of the 4-page manual (available at www.logangraphic.com) are simple instructions for that adjustment that should work well if you are willing to follow Step 1 and remove the sandpaper.

To see the Note click here.To hide the Note click here.
The full instructions are as follows:

Adjustment 45°

  1. Remove sand paper
  2. Place the 45˚ square flat against the wheel and up against bar (Fig. 7). Look for gaps against the bar.
  3. Adjust the bar using adjustment wrench until gap disappears.
Figure 7 of F200-2 Manual
Figure 7 of F200-2 Manual

When I don’t remove the sandpaper disk the technique doesn’t work as well, so I’ve come up with an alternate set of instructions:

  1. Put miter cuts on both ends of two long scrap pieces of your widest moulding.
  2. When you sand a piece of moulding, each end will use a different side of the sander. Call one side of the sander “A” and the other “B”. As you sand the two pieces of moulding, mark the back of each end of each piece with the side of the sander used (A or B).
  3. Find a good right angle, either in a reliable carpenter’s square or using other methods.
  4. Flip one of the pieces of scrap moulding upside down so you can join Corner “A” on both pieces to make a 90˚ (right) angle. Flipping is very important*.
  5. Put one piece of moulding along one edge of the reference angle (carpenter’s square) and slide the reference toward the second moulding until it just touches at one end or the other (if it touches at both ends, you are finished with Side A – skip ahead to Step 7). Measure the error gap (I like millimeters only because they are so small) at the end of the moulding that’s away from the reference line. Then measure the length of that piece of scrap moulding (using the same units of measurement).
  6. The adjustment screw on my sander had 32 threads per inch, and it was 109 millimeters from the pivot point. Based on that, your multiplier will be 25,000.
    To see the Note click here.To hide the Note click here.
    I’ll do the math just in case your sander has different measurements so that you can substitute the real numbers in for mine at the proper places. One complete turn of the adjustment screw is 360° or 1/32” and there are 25.4 millimeters per inch, so the constant multiplier would be 109 mm * 360° per turn of screw * 32 threads per inch ÷ 25.4 mm per inch ÷ 2 errors = 24,718.11. We’ll say 25,000.
    Divide the error you measured in the last step by the length of the moulding that you measured and multiply by 25,000. Your result will be the number of degrees you need to turn the adjustment screw. If the error gap was at the corner, then the angle is too large and you have to turn the screw counterclockwise to back it out. Conversely, if the error gap was at the end of the moulding, then the angle is too small and you have to turn the adjustment screw clockwise to push it out more. After making the adjustment, you may want to retest by repeating the process by starting at Step 2 and resanding the same two corners just used. Before sanding, I recommend drawing a line all the way across the end of the moulding with a pen or marker and then sanding until the line completely disappears.
  7. Repeat this whole process (starting at Step 4) for the other two miters, labeled “B”. Remember, for these two you will be playing with the adjustment screw on the other side of the sander.

That’s all there is to it. Congratulations.

To see the Note click here.To hide the Note click here.

Math Warning: a quick note about trigonometry (OPTIONAL)!

This process was concerned with angles, not distances, but since angles are harder to measure with any precision we had to convert. When you take the ratio of the two perpendicular sides (the sides that are 90° apart) of a right triangle containing the angle you are interested in, that’s called the tangent of that angle, and you can have a good calculator app on your phone find it for you (for my Droid, I found RealCalc Plus by Quartic Software at the Play Store and was happy to pay $3.50. There are plenty of other options, though).

The error angle you measured (indirectly) was actually twice as large as the real error. One problem is that the tangent curve is not generally a straight line, which means that the tangent of twice some angle is not the same as twice the tangent of that angle. That’s why the normal procedure would be to convert to angles, do the adding, subtracting, or multiplying, and then convert back to distances we can measure again. We were able to use the small angle exception, however. It turns out that for angles less than say 10°, the tangent curve IS pretty straight and the error introduced by taking our shortcut isn’t worth worrying about. That’s what we did, and that’s why the problem was so easy. That’s your math lesson of the day week month. Let’s get back to work.

Neutral Density Filters And Another “Brilliant Idea”?

Last updated on July 7th, 2019 at 08:58 am

A little over a year ago I was shopping for neutral density filters.  While doing my research, I triggered the following e-mail from the owner of Breakthrough Photography:

Hello Bruce,

My name is Graham and I’m the founder of Breakthrough Photography. I wanted to take a second to say hello and welcome you.

Seriously, on behalf of myself and the entire Breakthrough team I want you to know that we’re truly excited and grateful that you decided to join us.

Here’s what you can expect from me…

We publish new and actionable photography content to our blog 1-2 times a month.  You can see that I’m sending these emails from my personal address, so when you respond I’ll get back to you.

Sound fair? GOOD!

Now for the long exposure guide: click here to download our Essential Guide to Long Exposure Photography ebook. I wrote it as a reference guide so you can also save it to your iPhone or iPad for easy reference when shooting. Just click the link above on those devices. . . .

Best regards,


I downloaded and read his guide, and sometime during this research process I got another of my “brilliant” ideas and sent the following response to tell him about it:

To: Graham Clark (founder of Breakthrough Photography) at grahamclarkphoto.com
Sep 5, 2015 at 11:51 pm

Thank you for your Long Exposure Guide. I just spent a good part of the day shopping for neutral density filters for my wife, Nancy (She is a nature & wildlife photographer . . . . After seeing that your X3 line follows tradition and has only a 3-stop, 6-stop, and 10-stop version, I thought I should make a suggestion to improve versatility. . . .

Your last sentence in the “What Strength Should I Get?” section on page 14 [of the Long Exposure ebook] says “No need to buy a 10-stop!” That’s what I’m here to address. Instead of making a 10-stop just because everybody else does, you should make a 12-stop filter. Here’s why. Mathematically, I think your greatest versatility would come from having each filter in your line-up be double the previous one. From 3, you should go to 6 (as you did). From 6, the next logical step would be 12. With that combination of 3 filters, one could achieve every power (that’s a multiple of three (the lowest power sold)) in equal steps up to what would be the next step, 24. Obviously for 3 stops, grab the first filter. Similarly, grab the 6-stop for the next step. For 9 stops, the photographer should follow your advice and put the 3 and 6 together. Next, they would need the 12. 15 would be 12+3, 18=12+6, 21=12+6+3. The whole spectrum is evenly covered. If you wanted to go further, the 24 would be your next filter after the 12 (whether there is any market for the 24 is another matter). If the 3 is too powerful for some people (I don’t see why it would be), then the perfect choice would be a 1.5 (although at this point, the error of choosing either a 1 or a 2 ([so that] one wouldn’t need both), may not be significant enough to quibble about. You might say that this system unlocks the power of base 2. (Even though we use a base 10 number system, many of the things we use, like wrenches, measuring cups, and even cameras (I won’t even bother to mention computers) have been taking advantage of the benefits of base 2 for a long time). From a marketing perspective, I doubt anybody shopping for a 10-stop filter (because of tradition) would turn down the opportunity (and versatility) of a 12.

By the way, that base 2 thing reminds me of one . . . error I found at the top of that same section.  The first sentence of the second paragraph says “A 6-stop nearly doubles or triples the exposure time of what a 3-stop would have been . . .”  Since each f-stop lets in only half the light of the one before, technically, moving 3 stops would give one eighth (½ x ½ x ½) of the light and require a shutter speed eight times as long (instead of three).

To see the Note click here.To hide the Note click here.
Even though it has been over a year, the mistake still hasn’t been corrected.


He immediately replied

Sep 6, 2015 at 12:10 am

Hey Bruce,

Couldn’t agree more, 12 would be more logical given 3 and 6, however 10 is where the demand is.

“One should not create demand with a product, only channel existing demand onto a product.” Eugene Schwartz


While I was impressed with the quick response time, I would have preferred that he spend the extra time to think before he responded.  I sent him one more quick response,

To see the Note click here.To hide the Note click here.

Sep 6, 2015 at 1:29 am


Wow, thanks for responding. I understand that logic is not always the governing force in purchasing decisions . . ., but my suggestion does not contradict Eugene Schwartz’s advice (even though I think Steve Jobs probably never paid Mr. Schwartz much mind). You would be channeling the 10-stop market into the more favorable 12. Unless you are saying that all previous efforts to sell a 12-stop filter to those people have actually failed in the past. I didn’t realize anybody had ever tried. As I said in my last message, I didn’t think anybody shopping for a 10 would turn down a 12. For slowing down mid-day action (which is why I thought people bought the 10), I’ve seen some authors even suggest as high as 16. . . . Maybe you could sell all three (3, 6, & 12) as a package. . . . I just bought your 10 (possibly for the same reason everybody else did – because there aren’t any 12’s). . . .


but (as happens with most of my “brilliant” ideas) I never heard from him again.  And since we bought our 10-stop filter, we haven’t had the chance to use it (but I’m confident we will).

So, what do you think?  If you have experience with neutral density filters, especially experience that would be relevant to this controversy, I’d love to hear about it.  Would you prefer to have a 12-stop filter or a 10? Do you have any other insight into the use of this tool to promote better photography?

An Optical Illusion

Last updated on August 22nd, 2018 at 12:18 pm

When we do the gallery wrap on our canvas prints, I usually take just a tad along the edge of the image and digitally stretch it to several times its original width to wrap around the edge of the frame. When we do that stretching, we set up an optical illusion. When the image is hanging on the wall and you move to the side of the image, there will be some angle at which the stretched edge image will seem to be just a continuation of the image on the face of the frame. When I’ve discussed this with people in our booth at art festivals, I’ve left the calculation of that angle as an exercise for the listener.  Now I’ll describe the solution of that exercise in an article “Finding the Angle of the Illusion”.  This article is now on our website, but can’t yet be reached through the menu system. (I’ve recruited a friend to help me update the website, which is long overdue, so please bear with us).

Caution: this solution does involve basic trigonometry. Who would have thought we would ever use that stuff?

A Better New Way To Keep Your Mat Cutter Square

Last updated on August 22nd, 2018 at 12:20 pm

I have a faster and more accurate way to make sure my Logan Model 660 Framers Edge mat cutter is square that does not use a carpenter’s square. The procedure should be adaptable to other mat cutters also. I explain how in the article “Keeping Your Squaring Arm Square”, which is now on this website, but can’t yet be reached through the menu system. Enjoy!