# Weigh a scale with itself

Using nothing but a pocket scale and a flat surface such as a countertop, find the total mass of the scale, including the attached metal plate and the plastic tray/cover (left).

• If you press the TARE button, the scale's zero point will be set to whatever weight is currently on the scale.
• After using the TARE function, any weight placed on the scale will be displayed as a positive number and any weight removed from the scale will be displayed as a negative number.
• When you power on the scale with the ON/OFF button, it will wait for 3 seconds, then automatically trigger the TARE function.
• Values are displayed in grams.
• The scale's strain gauge is located between the metal plate and the body of the scale. The bar that the strain gauge is attached to has negligible mass.
• You may not use the UNIT or PCS functions.

• When sitting upright, the metal plate is the highest point on the scale. The metal plate does not come in contact with the plastic cover when the cover is on.
• The scale weighs less than the weight limit (300 grams).
• The center of mass in the X-Y plane is within the metal plate, as determined by this method — the scale will not tip if set upside down.

• Does the scale weigh less than 300 g? – msh210 May 15 at 8:55
• @msh210 Good question! It does. – Snowball May 15 at 11:48

You can:

1. Place the pocket scale upside down on the flat surface, turn it on, wait a bit, then turn it around. That should give you the weight of the body 1 (negative value), minus the weight of metal plate.
2. Place the pocket scale upside down sideways over the cover you showed, so that the metal plate hangs freely, but the body of scale is supported by the cover. (The metal plate does not come in contact with the plastic cover when the cover is on.) Turn it on, wait a bit, then turn it around. That should give you twice the weight of the metal plate (positive value). It's double since it first measures the plate hanging (and calls it zero), then it measures the plate on top. The difference will be twice the weight of the plate, which is what we want, since the plate was subtracted twice in step 1.
3. Press the "Tare" button, and put the cover on the weight. That should give you the weight of the cover (positive value).
4. Add the three numbers together to get the total weight of the scale.

Note: Corrected after comment by Jaap Scherphuis.

• Great solution. The scale is 104 g with the included batteries, or 90 g without them. – Snowball May 15 at 12:27
• Doesn't step 1 give you the the weight of the body minus the weight of the plate? (i.e. not the weight of the body alone, which is the weight of the whole scale minus the plate). So later in step 4 you should add twice the weight of the plate back on. – Jaap Scherphuis May 15 at 12:31
• @JaapScherphuis of course, you're right... Edited and credited :) – CG. May 15 at 12:52
• @TCooper With this scale, it doesn't fail because the center of mass in the X-Y plane is within the metal plate. I'll add that information to the question. Thanks for pointing out the omission! – Snowball May 15 at 20:05
• Gotta say, this was quite a fun little puzzle! – tgm1024--Monica was mistreated May 15 at 23:37

Hold the scale in the air by the metal plate, so that the base of the scale is pulling down and creating tension on the strain gauge equal to the base's weight. Turn the scale on and let it tare. Now place the scale on the table. Relative to the previous tare, this new measurement no longer has the base pulling down (so +weight of base), and it also now has the weight of the plate on top (so +weight of plate). This means the displayed measurement is weight of base + plate, i.e. weight of the scale.

• That's a clever solution, but I'm not sure it would work in practice. I believe you would have to keep incredibly steady. D=0.01 grams, which is very very little... Note: I'm not sure, it could work. – CG. May 16 at 14:30
• @Snowball, can you try this out and let us know how it compares to the accepted answer? – knrumsey May 16 at 14:59
• @knrumsey: This was a bit tricky to pull off, but it works! CG.'s solution gave 104.165±0.004 g (SEM, N=10) and hpp3's solution gave 104.163±0.027 g (SEM, N=10). – Snowball May 16 at 19:21

Can you

power it on, put it upside down (with the plastic cover on top), wait for 3 seconds, and then put it upright again?

If

it says, for instance, -500 grams, the scale weighs 500 grams?

• this is what I thought too, but I think it should not be that obvious :D when putting upside down might be making scale touches on the ground from other points? that information is not given though. – Oray May 15 at 8:09
• @Glorfindel exactly what I thought! – William Pennanti May 15 at 8:14
• Not quite, that wouldn't include the mass of the metal plate. – Snowball May 15 at 8:21
• @Oray: The metal plate is the highest point on the scale. Thanks for pointing out that omission; I'll add it to the details in the question. – Snowball May 15 at 8:28

I'd like to examine the pieces of the puzzle so that we can build to a solution. The assumption going forward is that each "setup" is stable such that nothing is moving or being held in place by anything other than the weight of the pieces. Also, that the scale isn't sideways, as it's probably not designed to handle that.

What's the meaning of the number that appears on the screen? It's not necessarily a reak weight, but the difference in weight between the current setup and a reference setup - thus the need to use the TARE function.

So then, how do different setups affect the measurement? Each relative number measures the compression on the contact point (i.e., strain gauge) between the plate and the base. If the plate and base are being pushed together, the pieces sitting on top of the contact point are measured. But if the plate and base are being pulled apart, the pieces hanging from the contact point are measured as negative values.

How can we use this? There are a handful of meaningful setups we can use as our reference points. From there, we can place the scale in setup1, use TARE, move it to setup2, and record setup2 - setup1 = number_on_screen. With a couple of those in hand, it's simple to solve for each unknown.

Here are some relevant setups (the list may not be exhaustive, but is sufficient):

1. Scale right-side-up, lid to the side: measures plate
2. Scale right-side-up, lid on plate: measures plate + lid
3. Scale up-side-down, lid to the side: measures base
4. Scale up-side-down, lid on base: measures base + lid
5. Scale, with lid attached, up-side-down (the plate hangs): measures plate, but negative

Finally, we can combine each of those in pairs, record the value displayed, and form equations. Ideally we can isolate a piece so that we can get its weight directly. If/once we have one weight, we can discount it as a "variable" and solve for another weight in a shared equation. Fortunately, we can isolate the lid and the plate as follows: 1. TARE for setup 2, then setup 1 => X = (plate + lid) - (plate) = lid 2. TARE for setup 5, then setup 1 => Y = (plate) - (-plate) = 2*plate

So we have the weight of the lid and the plate, but need to get the base. We need any setup that has the base as well as any setup that does not have the base. The simplest is: 3. Tare for setup 1, then setup 3 => Z = (base) - (plate)

Finally, with some algebra, total = (lid) + (plate) + (base) = (X) + (Y/2) + (Z + Y/2).