When a metal ring is heated, does the hole expand, contract, or stay the same size?

Spoiler alert: The puzzle is answered in the update at the end of this post.

Most solids expand when heated and contract when cooled. Water/ice is anomalous in that it expands when cooled, at least near its freezing point. If you’ve been unfortunate to forget a bottle of water in the freezer, only to find that it has broken once its contents have frozen, then you’ve seen this effect first-hand. Similarly, if we don’t clear the water from our outdoor pipes in winter, we run the risk that they might burst should the water within freeze. You can also observe this effect by carefully filling an ice-tray up to a mark, then noticing that upon freezing the ice has expanded so that it is now higher than the mark.

Why does water have this unusual behaviour? Its molecular structure and hydrogen bonding tell the story.

Ordinary solids (that is, virtually everything except water/ice) expand upon heating and contract upon cooling. Molecules jiggle about, and when a substance is heated the jiggling becomes more vigorous on average; just as dancers move apart on a dance floor when the dance becomes more energetic, molecules tend to move further apart when their average energy increases due to heating. The attractive forces between the molecules keep them from flying apart unless the jiggling becomes truly enormous, which requires correspondingly enormous heating; we call this vaporization.

The rate at which solids expand when heated depends on the substance. Metals tend to have higher rates of expansion (per degree change in temperature) than non-metal solids, but there is variation even among metals. A table of expansion coefficients can be found here or here. As you might expect, liquids have a much greater rate of expansion than solids (because the attractive inter-molecular forces are smaller for liquids than solids), and very hard solids (such as diamond) tend to have smaller expansion rates, thanks to their very strong inter-molecular forces.

This phenomenon has practical implications. Engineers design large structures with expansion joints to prevent buckling and cracking during summer heating. You can see such expansion joints in railroad tracks, bridges, highways, and even sidewalks. You can imagine that in situations where temperature changes are extreme (for example, in spacecraft, or supersonic jets (see also here)), research into alloys that have minimal temperature expansion is important.

So here is the puzzle: Consider a washer, or some other metal ring or disk with a hole in it. When the ring is heated, we expect the ring to expand, and experiments confirm that it does expand. But does the hole in the ring expand, contract, or stay the same size?

This is meant as a training exercise, so give the puzzle serious thought before looking up the discussion (which is all over the internet). Just knowing the answer to the question is useful, of course, but the struggle to think through the puzzle is far more useful, in the same way that physical exercise is much more useful than watching others do it!

PS: The puzzle can be found on Page 221 of a delightful book by Lewis Carroll Epstein, called Thinking Physics. Full of insight, it is highly recommended for young physics students.

PPS: The dancing analogy is meant to be taken loosely; molecules do not move apart because of a conscious decision to avoid striking each other, the way polite dancers would. Rather, their average speeds increase thanks to their increase in energy, which allows them to move further away from each other before they are reeled  back in by the attractive forces acting between them.

Update: I’ve had so many hits on this post that it occurs to me that the solution to the puzzle might be useful, so here it is.

The most important point that should be made is this: The only scientifically valid solution to the puzzle is to go out and do the experiment! No amount of reasoning can ever convince us of anything in science; of course, we value reasoning, and use it to guide our thinking; but ultimately, one must do the experiment.

So go, do the experiment!

But don’t throw your wedding ring into the fire! Rather, think about what you do when you are trying to open a Mason jar, and the screw-top metal lid is stuck. You either tap on the lid with a spoon (to try to jar loose any part of the lid that is stuck), or you place the lid under hot water. You do the latter because you know the metal lid will expand more than the glass jar, and so it will be easier to get the lid off.

And by saying the metal lid will expand more than the glass jar, what we really mean is that the hole in the lid will expand.

And that is the end of the story. You do the experiment, repeat it in many different circumstances, and you draw your conclusion.

But nevertheless, it is helpful to try to use reason to understand the situation, as this will help us understand such phenomena. There are a number of ways to reason your way through to the correct conclusion, but Epstein’s answer from page 222 of Thinking Physics is a good one, and I paraphrase it in the following paragraph.

He suggests taking a square piece of metal plate, dividing it with a 3-by-3 grid into nine equal smaller squares. Then heat the entire plate. Each of the smaller squares expands. But if the central square were missing from the start (a hole), then the same expansion would take place in the other 8 squares, leaving a bigger hole. Alternatively, if you heated the entire plate and then removed the central square at the end, after it has expanded, the remaining hole is larger than the original size of a small square.

Similar reasoning applies no matter what the shape of the original metal ring is.

If you feel you’ve understood this, test your reasoning on Epstein’s following puzzle, taken from page 222 of his book: A nut is very tight on a screw. Which of the following is most likely to free it? Cooling it, heating it, either, or neither?

And, once again, the best test of your reasoning is to go try it!

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About Santo D'Agostino

I am a former university prof, having taught mathematics and physics since the mid 1980s. I have also been a textbook writer/editor since then. Currently I am working independently on a number of writing and education projects. I love math and physics, and love teaching and writing about them. My blog also discusses education, science, environment, etc. http://qedinsight.wordpress.com Further resources, and online tutoring, can be found at my other site http://www.qedinfinity.com
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36 Responses to When a metal ring is heated, does the hole expand, contract, or stay the same size?

  1. Someone says:

    WHY COULDN’T YOU PUT THE ANSWER AT THE TOP OF THE PAGE????

    • The first line in the post invites you to scroll down to the “Update” towards the end of the post to learn the answer to the puzzle. The answer is placed at the end to encourage readers to think about the puzzle, and perhaps make observations or otherwise play with this situation, before reading the answer.

  2. J says:

    Running a jar under hot water also heats the contents increasing the internal pressure. Bad example.

    • I have done this many, many times, and can assure you that the increase in temperature (and therefore pressure) of the contents of the jar is neglible. It is typically enough to place the lid under hot running water for less than a minute, being careful to angle the jar so that the hot water runs off the lid into the sink, and doesn’t run down the jar. Because the specific heat capacity of the metal lid is high, and its mass is low, it expands sufficiently in a short time. The mass of the contents of the jar is much greater than the mass of the lid, so it would take much longer for its temperature to change by a significant amount.

      But if you don’t like this example, any of the others serves to make the point.

  3. DG says:

    A jar lid is not the a ring, it’s a solid piece. The flat surface would expand outward too.

    I’ve seen an experiment done with an actual ring and the hole got smaller. The reason is that the metal expands in all directions, inward and outward. It’s analogous to a pressure cuff on your arm. The cuff is put on your arm and then air is forced into the cuff. The cuff expands in all directions, inward and outward.

    However, it just occurred to me that the thickness of the ring may have an effect on what happens. A ring with very thin wall may not behave the same way as a ring that is much thicker.

    Also, you can imagine the metal plate. If you cut it tangentally to the hole in the center and then heated it up, would the cut edge expand? Then if you instead cut it in a U shape, would you expect it to only expand outward from the center of the U, or outward from the center of the metal?

    • “A jar lid is not the a ring, it’s a solid piece. The flat surface would expand outward too.”

      I am thinking of a Mason jar, where the lid and ring are separate. But I don’t think there is a substantial difference.

      “The reason is that the metal expands in all directions, inward and outward. It’s analogous to a pressure cuff on your arm. The cuff is put on your arm and then air is forced into the cuff. The cuff expands in all directions, inward and outward.”

      It’s not obvious that a gas system should behave analogously to a solid. For example, the pressure, temperature, and volume of a gas are described by an equation of state, but the same equation does not apply to a solid. I argue towards the end of this comment that the behaviour is quite different.

      “I’ve seen an experiment done with an actual ring and the hole got smaller.”

      Ultimately, experiments are definitive. I can reason and argue as best I can, but nobody should believe me; rather, one should do the experiment. Having done the experiment with the rings on my Mason jars, I have experienced the results myself. However, I’m always interested to see other experiments. Have you seen such an experiment (where the hole got smaller) online? Or do you mean that you physically observed the experiment as it took place? If the former, could you please send me a link; if the latter, could you explain the experiment in a bit more detail?

      “However, it just occurred to me that the thickness of the ring may have an effect on what happens. A ring with very thin wall may not behave the same way as a ring that is much thicker.”

      I don’t see why the thickness of the ring would matter. It might be interesting to think about a ring that is one atom thick. Imagine a simple model of the ring, where each atom is represented by a tiny ball, and each ball is connected to its nearest neighbour by a spring. The balls are all vibrating, and the vibrational amplitude increases as the temperature increases. As the temperature increases, the increased vibration makes the ring larger, and so the hole in the centre also gets larger. A real three-dimensional ring is much more complicated, but the thin ring is suggestive.

      “Also, you can imagine the metal plate. If you cut it tangentally to the hole in the center and then heated it up, would the cut edge expand? Then if you instead cut it in a U shape, would you expect it to only expand outward from the center of the U, or outward from the center of the metal?”

      I’m not sure I understand where the cuts are, so I’m not sure the point you’re making. (I’m guessing you might mean “radial” instead of “tangential.”) But let me try to respond to your last paragraph with this argument (this is an argument that the hole in the ring should get larger when the ring is heated). Imagine cutting a ring along a diameter, so that you get two approximate half-circle rings with two small gaps in them. To be a bit more definite, suppose that the ring is the region of the xy-plane between the circles x^2 + y^2 = 1 and x^2 + y^2 = 2, and suppose that the small gaps have been cut at x = 0, one for y > 0, and the other for y < 0. When you heat the half-rings, the small gaps will definitely get smaller, as the metal "expands in all directions." Now imagine a ring with no gaps; there would be the same tendency to expand in all directions, but because there are no gaps, and because metal has considerable rigidity, this tendency to expand in all directions actually forces the left and right parts of the ring apart (in lieu of expanding into non-existent gaps). By symmetry, the expansion is in all radial directions, and so the hole in the centre gets larger.

      This argument highlights the difference between a gas system, such as the blood pressure cuff, and a solid system. The rigidity of the metal forces the expansion to make the hole larger. However, the blood pressure cuff is so flexible that it allows the air inside to expand into the “hole” as the presssure of the gas increases.

      Does this respond adequately to your last paragraph? It may be helpful to the discussion to sketch the figures you talk about and then photocopy them using a “zoom” option. After seeing what happens to the gaps and holes when the figures are zoomed, the questions we are discussing can then be framed as, “How is the atual metal ring expanding the same as the zoomed copied image, and how are they different?” That might be a useful way to continue the discussion, and might also be a useful way to explore the questions in your last paragraph.

      Thanks for the very thoughtful comment, and all the best wishes!

  4. aa says:

    Hi… I was wondering … We know that if you heat a steel ball, it will expand based on its coefficient of thermal expansion. What if we freeze or cool the steel ball (at room temp. initially) … is it going to shrink?
    thanks,

  5. aa says:

    can you please elaborate on this? I know that there is a value for thermal expansions,,,but never heard of thermal shrinking or contraction!

    the reason I am asking is that I read about particle bed of solid material (steel for example) and it says that when heat is applied to the bed, the bed expand due to thermal expansion of the materials. Then, when the heat is off, the particles cool down (room temperature). This cause settling (re-arrangement) for the particles and the bed consolidate.

    if we would freeze the particles instead of heating them, would we get the same effect?

    thanks,

    • Yes, indeed, virtually all solids contract upon cooling and expand upon heating. Imagine a solid as a collection of vibrating balls (the atoms) connected by springs (the inter-atomic forces); a simplistic model, but it may help you to understand the situation. When the solid is heated, the balls have more energy and thus can stretch the springs further as they vibrate; this causes the entire solid to expand, because the spaces between the balls has increased. On the other hand, if the solid is cooled, the balls have less energy, and so the springs pull them more closely together; this causes the solid to contract, as the spaces between the balls has decreased.

      There are some notable exceptions to this almost universal phenomenon. One exception is the behaviour of liquid water *as it freezes* to form solid ice. In liquid form, the water molecules slide around each other in relatively close packing. However, once the water freezes, the molecules are “pushed out” into a hexagonal crystal structure, with a lot more space between each molecule. You can observe this phenomenon by freezing ice in an ice tray in your refrigerator. Carefully observe the water level before freezing, and then observe the position of the top surface of the ice after it has frozen, and you will see that the ice has been pushed up higher than the position of the upper surface of the water before freezing.

      However, one the ice has already fully formed, then further decreases in temperature will cause the ice to contract, in the same way as virtually all solids. If the ice is then warmed up, it expands, in the same way as virtually all solids do, until it reaches the melting point. It is just at the point of freezing or melting that water/ice behaves strangely.

  6. Bill says:

    OK, I’ve been noodling on the last question (the bolt and nut one) all day and ended up finding this post. I started the day convinced that heating the bolt makes the nut internal diameter bigger faster than it makes the bolt diameter bigger. I even had a great reason worked out. But after grinding through a bunch of math I have now convinced myself that the OD of the bolt will grow exactly as fast as the ID of the nut, meaning no net difference (if the nut and bolt materials are the same). The mason jar example works only because the glass has a much lower coefficient of expansion than the steel lid does.

    Can you confirm or deny? :)

    • Heating the nut and bolt will help to loosen it, cooling it will help to make it tighter.

      To see this, the best way is to actually do the experiment. (I am told that people do heat up rusty nuts and bolts in an attempt to loosen them, so it seems to be an accepted trick of the trade in some circles.)

      To reason about this, consider the following argument. Imagine taking a circular metal plate and drawing a bunch of concentric circles on it. When the plate is heated, all of the linear dimensions increase, including the distances between the concentric circles. Now imagine that the inside diameter of the nut is (sort of) one of the concentric circles, and the outside diameter of the bolt is (a smaller) concentric circle. The distance between these two circles is the gap between the bolt and the nut. Upon heating, this gap will increase in size, just as all of the other linear dimensions in the system. Upon cooling, the size of the gap will decrease.

      • Bill says:

        That’s what I thought also, then I fouled everything up by trying to do the actual math. If you calculate the linear expansion in length based on the circumference of the ring, and use that to back calculate the new diameter of the ring, you get value X. If you calculate the linear expansion a plain disc inside that ring (just based on diameter, not circumference), you get…. the same value of X. Unless my math is wrong.

        In many ill concieved rescues of old lousy motorcycles I have taken, I have found heat to be a huge help. But i think that may be because aluminum (which is often what the bolt is stuck into) expands faster than steel (which is what the bolt is).

      • If I understand you correctly, then your math is correct. But if you think of the nut and bolt as being separated by a small gap (too small, because there is a bit of rust or dirt that is clogging the gap), then there are two circles, with two different radii involved. The smaller circle you can think of (roughly) as the outer radius of the bolt, and the larger circle you can think of (roughly) as the inner radius of the nut. Specify the two starting radii, and determine the difference between the two, which I’ll call Y. Then expand each of the inner and outer radii by the same factor (assuming they are made of the same material and have the same temperature change); suppose the expansion factor is A. Then calculate the new gap between the two radii; the new gap should by AY. Thus, the size of the small gap between the bolt and nut has increased slightly, enough (one hopes) to reduce the friction between them, allowing us to unscrew.

        I don’t have a lot of experience messing about with old motorcycles (mainly this past winter when I helped my son to attempt to rehabilitate an old Honda CBR), so I’m happy to hear about your experience with steel and aluminum, which I’ll keep in mind when this comes up again.

  7. Bill says:

    In other words, “Mind the gap”. :) That’s that I was missing, thank you for taking the time to explain it. I’ll work the math for a real world example, a steel 6 mm bolt shaft in a steel hole.

    Neat! The math works now. A 6mm bolt shaft in a 6.05mm hole at 60 deg F (15 deg C) has a .05mm gap. Heat both the bolt and the hole to 300 deg F (148 deg C). The 6mm bolt is now 6.0127968 mm. The 6.05mm hole is now 6.062903440. The gap is now .050106640.

    So the gap did indeed grow by .0001mm as a result of heating.

    Other factors from heating are probably more relevant in terms of why the bolt comes out easier after a few heat cycles… for example the bolt will heat and grow first, then the hole, which will break up stuck stuff. And if you hit the bolt with penetrating oil after heating both it is shrinking faster than the hole, and getting oil in there, etc.

    Thanks again! A fun thought exercise. If you have any problems with the CBR feel free to email and I’ll help how I can.

    • I bet your comments about the reason why the bolt comes out easier upon heating are correct. It could be the same with the jar lids, which may come unstuck partly because the hot water gets under the lid and dissolves whatever is causing them to stick. (My mother used to tap the lid with a heavy spoon, which may do the same job of jarring loose whatever is stuch.) Real life is a lot more complex than idealized textbook situations.

      Thanks, I may take you up on the motorcycle consultation some day!

  8. Peter Brown says:

    Any wheelwright over the last several hundred years will tell you that as a steel tyre is heated, it expands and the internal diameter gets bigger. The wheel rim is made of several sections (arcs) of timber fitted loose onto the timber spokes and the spokes into the hub. the metal tyre is made so that it is slightly smaller than the circumference of the wheel and is heated to red-hot. It expands until it is larger than the wheel rim, then placed in position and cold water poured onto the hole. This has the effect of putting out the inevitable fire and shrinking the tyre onto the wheel. This process has the effect of tightening all the joints in the wheel together and gives the characteristic hub moving outwards away from the plane of the wheel.

    • Thank you for a wonderful contribution.

      My father-in-law is a carpenter (in his 80s now), and he learned the trade from his father, who specialized in making carts and cart wheels. He reports they used the same method you describe so vividly.

      It’s too bad that I didn’t think of including my father-in-law’s experience in the original post, SO I’m particularly grateful that you posted your story.

      Thanks again!

      • Peter Brown says:

        It is just a guess, but my theory for why the expansion of metal in a wheel tyre is not towards the centre of the circle is that the tyre is made from a strip of iron formed into a circle and the ends welded together. When heated, the outer surfaces of the metal may be fractionally cooler on contact with air but the centre of the metal retains the heat longer and it expands along it’s length a little quicker than outwards to it’s exposed surfaces. This linear expansion inevitably makes the tyre circumference larger

  9. Randy says:

    I came across this post while looking to understand metal expansion to try and get my car repaired. I am trying to slide a bearing onto a shaft and while I expect some resistance I seem to have encountered what I consider excessive resistance. If I understand this if I were to leave the inner shaft in the cold, 30f (-1c) and bring the outer bearing to room temp the shaft would be smaller diameter and the bearing a larger one?

    While my ultimate goal was to fix my car I have to say I found this post extremely interesting. Science in a textbook is mildly interesting to me but science in the real world grabs me. I’m a hands on guy so the math will be much less interesting that the real world act of the parts on my car fitting. Thanks much I think I learned something today!!!

    Science is fun!!

    • That’s right, keep the shaft in the cold, and put the bearing where it is warm, then bring them together and (one hopes) they will fit together before their temperatures equalize.

      The expansion/contraction effect is very small for the kinds of temperature differences that it is easy to generate around the home. Of course, one can always throw the bearing into a blazing fire, but that may be impractical because of the potential damage to the bearing.

      I’m glad you enjoyed the post; science is indeed fun!

      Let me know how things work out, and whether the bearing finds its way back onto the shaft.

  10. Pingback: Problem 27: When a Metal Ring is Heated, Does the Hole Expand, Contract, or Stay the Same Size? (23 February 2013) | QED Infinity

  11. Science student says:

    How can you be sure that when a solid is heated and expands in size, the increase in size is not caused by more atoms being added?

  12. Science student says:

    Yes, I do understand the kinetic theory of matter etc. but I need to think of an expirement to prove that no more particles are added and I am a bit stuck.

    • Hi Flutediva,

      Yes, it’s a very good question. Why not measure the mass of an object when it is at different temperatures? If you can manage it, it would be good to set up your apparatus so that the object is on the mass scale for the entire experiment, as it is gradually heated or cooled. Noting that the mass reading does not change as the object is heated and cooled would provide evidence that heating does not add matter and cooling does not remove it.

  13. Cal says:

    I’ve seen on a national geographic show of a Ferarri factory where they cool a metal ring in liquid nitrogen, then place it on a stud where it is meant to stay, then as it warms to room temperature, it becomes stuck on the stud. I’ve also seen this in person at a mining equipment factory. This would indicate that the inner diameter of that particular ring is larger when it is cold, and smaller when it is hot. I would imagine that the outside diameter of the ring is SMALLER when it is cold, and LARGER when it is hot. The reason for this is, if you imagine taking that ring and slicing one quarter or one eighth or one whatever of it out, that piece would expand and contract just like a solid piece of metal. You have to realize that there is no metal in the inside of the ring affecting its size. Just air. The only material we are concerned with is in the body of the ring. Those would be the forces acting in and out from the center of the ring. But we also need to take into account the forces acting tangentialy on the material. now we have to imagine the ring as being unfurled into a straight rod. The length of the rod will decrease as it cools and the length will increase as it heats. That would translate to the circumference of the ring. This effect will affect the shape of the ring but so will the in and out forces. I think it is just a mixture of all these things and every piece will act differently. Its a fight between the in and out forces and the tangential forces. If you had a mile wide ring with a wall thickness of just 1mm, the tangential forces would have much more effect than the in out forces and the inside and outside diamter would get smaller as it was heated. But if you had a 5 inch wide ring with a wall thickness of 1inch, the in out forces would have more of an effect. The inside diameter would get bigger, but the outside diameter would get smaller as it cooled. I think Its a case by case basis.

  14. Allan Towe says:

    A steel rifle with steel bases and rings holding an aluminum scope vs.a steel rifle ,aluminum bases,aluminum rings,aluminum scope…….which makes more sense to you in a temp range from 10 to 90 degrees Farenheit…? Thanks,
    Allan Towe

  15. Peter Brown says:

    I have no idea how to start a new topic but I have been curious about something for a long time. In an electric kettle, as soon as the water begins to heat, there is a noise. However, just before the water comes to the boil, it goes very quiet followed soon after by the noise of the water boiling. Does anyone have any ideas as to why the kettle goes very quiet at this point?

  16. Teng Guo says:

    I am a mechanical engineering graduate and recently I’ve been re-visiting some of my old stuff. I would like to share my explanation, from a different perspective, on the metal ring expansion subject. So again the conclusion is the hole will expand, and here’s why:
    First let’s look at thermal expansion at atom level: metals are formed by attraction forces between electrons and nucleus. The magnitude of this attraction force determines how closely the atoms are packed together. So the stronger the attraction, the shorter the spacing between adjacent atoms. And we know that the mobility of electrons depends on temperature. When temperature increases, electrons are more free to move, as a result, the attraction or bond between electrons and nucleus becomes weaker. This means the atoms that form the metal space out when heated, which is why we see an expansion of the metal piece at the macroscopic level.
    Back to the problem of heating a metal ring, when we heat the ring, indeed the atoms ‘want’ to space out in all directions. But there are favored and unfavored directions. Let’s look at the layers of atoms that form the inside surface of the ring (basically the hole-side material). If the hole was to shrink upon heating, these atoms would be placed closer towards the hole center. However, we know as we get closer to the center, the circumference of the hole decreases, which means a reduction of spacing between atoms circumferentially. This is contrary to what the atoms would ‘like’ to behave and thus I call it unfavored direction.
    In summary, thermal expansion from the microscopic level is just a spacing-out behavior of the atoms. This spacing-out happens in all directions but for the case of a ring, there is favored directions: basically any directions pointing away from the center and ‘out’, as the phrase ‘space out’ indicates.

  17. CW says:

    From an Electric Motor Guy.
    Hot dropping bearings on shafts is a common industrial assembly technique. The bearing is heated, often with an induction heater (the shaft remains at room temp). The ID of the bearing increases enough to slip on the shaft. When the bearing cools to room temp you have an interference fit that is very strong with no mechanical force/damage applied to the bearing.

    Heating stuck bolt/nut assemblies is also a common technique. I have usually done this with a torch. Part of the “trick” is to apply the heat/flame directly to the nut, so it experiences a greater temperature increase than the threads on the bolt. You have to work fast.

  18. scyther says:

    Great thread – I came upon because I have a shank of a bit stuck in the chuck of a mortiser – severely stuck. Even heating did not do the trick, though I may not have heated it enough. Just wanted to verify that the hole in a donut of steel does indeed get larger upon heating.

    • Yes, indeed, the hole in the donut of steel does get larger upon heating.

      • scyther says:

        Thanks for the reply, Santo. What goes in can come out; the solution will require finding a way to use as much mechanical advantage as was applied in forcing the shank into the chuck; the machine is built to apply force in that direction – none at all in the other. A lesson in learned in the very tiny tolerances in machined steel parts!

  19. scyther says:

    just to throw in another real-world twist, I am loathe to heat the chuck a great deal lest the bearings get damaged by distortion.

    • You certainly have a challenging problem on your hands, with a metal object stuck inside another metal object. I’m afraid this is far outside my practical experience, as I have never used a mortiser. If you are able to solve the problem I would be happy to hear out it turns out.

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