# How to guess the content of a christmas present?

Let us assume that the present does not make any recognizable sounds when shaken (meow splat - the present now contains a dead kitten). Let us furthermore assume that the internal state of the present does not change, so that any measurements can be repeated.

I will consider it cheating to unwrap the present, or shine any form of light, rays or particles through the present. Not many people has x-ray or neutron scattering devises available anyway. - and obviously it is cheating to squeeze the present.

What can be measured?

• Size and shape of wrapping(hey mom ... my wish elephant will not fit in here)
• The mass
• Position of center of mass
• Moment of inertia tensor
• Vibrational resonance
• Magnetic resonance

If the seven scalars in mass, position of center of mass and moment of inertia tensor were measured, what could be said about the shape and mass distribution of the present?

If vibrational or magnetic resonance were measured, what could be said about the shape and materials of the present?

Would the thermal radiation not just depend on the temperature of the wrapping? what if the room temperature were dropped, then the present would heat the wrapping depending on the heat capacity, and thermal conductance. -but can it be used for anything?

This question has been heavily edited to correct my previously mistakes, and reflect my progress. THANK YOU for the answers already given.

-
Engineering answer: open it. –  Colin K Dec 31 '11 at 15:11
Mass and dimensions alone let you verify against shipping information. If you already have a conjecture as to the nature of the gift, that’s plenty. –  Jon Purdy Dec 31 '11 at 22:49
Social Engineering answer: threaten the giver by calling the bomb squad if they don't talk. –  metzgeer Jan 1 '12 at 5:45

Rotating about an axis not going through the center of mass does not give anything new. There is Huygens-Steiner theorem saying that moment of inertia with respect to any axis is $$I' = I_\text{c.m.} + m d^2$$ where $I_\text{c.m.}$ is the moment of inertia about the center of mass and $d$ is the distance between new axis and the center of mass (the axes must be parallel).

The general expression for the moment of inertia tensor about a displaced axis can be found here.

Hence the measurements of the moment of inertia tensor about multiple axes will give only the tensor itself and position of the center of mass. A solid brick of appropriate shape placed in the center of mass will give the same result.

-
Only true for solid objects though, you could tell if it was a bottle of liquid rather than a block of glass. You should also be able to measure the viscosity - at least enough to know it's a bottle of wine or cough syrup. –  Martin Beckett Dec 31 '11 at 19:25
I now see that you are right, the off axis experiment gives nothing new, and indeed the moment of inertia is just one tensor, and not infinite many scalars. –  Hans-Peter E. Kristiansen Jan 2 '12 at 11:58

Everything you can do (other than “shaking” — mechanical interactions) is going to involve “light, rays or particles”, because one of the principles of physics is locality; any information you get about the inside of the package must be from something that propagated out of it. Nearly anything you can do is going to involve “shining light”, that is, photons (= electromagnetic waves), though they may be extremely low frequency.

• You can in principle measure the gravitational field produced by the contents of the present, and thereby obtain more information about the mass distribution. However, this is an extremely weak effect and entirely impractical; its only advantage is that it definitely does not involve any photons (but it would involve gravitons, if they exist).

• Change the ambient temperature and observe the pattern of thermal radiation emitted by the package, using a thermal camera. This is “shining light” but only light that is already going to be there.

• Place the package in an electric or magnetic field and observe the forces produced on it and how it alters the field. The refined form of this is NMR/MRI imaging; however, that involves some high-frequency electromagnetic waves. If you want to stick to low frequencies and everyday equipment, you could move a magnet around the package and note any attraction. (Or rather, for sensitivity, hang the magnet on a string and move the package next to it.)

• Shake it better: aim a speaker at it, and examine the frequency analysis of the response (compared to the sound without the package present). This can tell you the resonant frequency of shapes inside the present and therefore some guess as to, e.g., the size of flat surfaces. The refined form of this is ultrasound imaging.

-
Well the measurement of the gravitational field would surely just give the same as a solid mass at the center and hence not tell anything new. –  Hans-Peter E. Kristiansen Jan 2 '12 at 12:00
I would not personally consider measuring the thermal radiation cheating, and measuring both NMR and resonances is brilliant ideas, but would one be able to tell anything objectively from the measurements? - size, shape, material,... –  Hans-Peter E. Kristiansen Jan 2 '12 at 12:05
(1) No, you can get information about the mass distribution from the gravitational field; consider the mapping of lunar mascons, which according to that article was done just by analyzing the trajectory of a probe. (2) MRI gets you a 3D image, i.e. size and shape of different regions, and apparently some information about composition as well. (I don't have any background in these topics so I'm not providing much detail; consider this to be existence, rather than constructive, proof, if you will.) –  Kevin Reid Jan 2 '12 at 16:31
I would just open it –  Awesome Mar 7 at 16:43

The measurements which you have allowed only support the determination of a very limited number of variables.

Note that the moment of inertia w.r.t. axis parallel to vector n=[n1, n2, n3] equals

$$I_n= n^{T}In=\sum_{i=1}^3 \sum_{j=1}^3 I_{ij} n_i n_j$$

where In is the moment of inertia w.r.t. axis n, I is the inertia tensor and Iij is its corresponding matrix element.

Thus, no matter how many measurements of moment of inertia w.r.t. to different axes one performs, all these measurements yield just the inertia tensor, which is merely 9 numbers. Also, it can be shown that I can be represented by a symmetric positive semi-definite matrix. Spectral theorem for symmetric positive semi-definite matrices ensures the existent of a basis in which the matrix is diagonal:

$$I = \left[ \begin{array}{ccc} I_x & 0 & 0 \newline 0 & I_y & 0 \newline 0 & 0 & I_z \end{array} \right]$$

where Iv represents the moment of inertia w.r.t. axis v and x, y, z is the basis composed of eigenvectors of I (in the context of inertia tensor x, y, z are also called principle axes of I).

This means that all the measurements of moments of inertia w.r.t. different axes yield information that can be represented with just 3 (real non-negative) numbers. We disregard the information about the orientation of the principle axes here assuming that presents that can be transformed into one another via rotations can be identified. As for presents which are each others' mirror reflections, they cannot be distinguished by measuring moment of inertia.

The measurement of center of mass yields additional 3 numbers: the coordinates of the center of mass. The measurement of the mass, obviously yields just one more number. Thus all the allowed measurements yield a total of 7 numbers.

It is not difficult to imagine presents whose structure cannot be encoded on 7 numbers alone. All the presents most children would want have this property. Thus, the allowed measurements cannot tell many of these presents apart.

Measurements of moment of inertia with respect to parallel axes don't give much more information due to parellel axis theorem.

-
Ahhh yes I see. The moment of inertia is indeed just one tensor, and not infinitely many scalars. If I took a random object, did the measurements, send you the numbers, would you theoretically be able to build(form in clay, carve in wood) an object with these properties that looked completely different? can the full set of objects with these properties be described in a more intuitively way(don't just give me the same seven numbers:o) ? –  Hans-Peter E. Kristiansen Jan 2 '12 at 12:18
Good question. Think of an object built from three very thin, light and long sticks intersecting at right angle in their middle. Place a mass at the intersection point (this is our knob for controlling total mass of the object independently of other variables). Place 6 additional masses at a distance from the intersection point, one on each side of each stick. You have 7 masses which allow you to control the total mass, center of mass and tensor of inertia. –  Adam Zalcman Jan 2 '12 at 12:36
Yes, I see that is one way, but it does not bring us closer to a description of the full set. ... and will it fit in the volume given by the wrapping:o) –  Hans-Peter E. Kristiansen Jan 2 '12 at 12:41
Oh, you mean the set of objects with the same 7 numbers? You can define such set by the type of transformations which keep the 7 numbers unchanged. If you pick a straight line going through the center of mass then you can move two small pieces of the objects mass along the line without changing the total mass or CoM. Doing this along three appropriately selected axes allows you to keep tensor of inertia unchanged. This way, you can make any object eventually resemble a hedgehog. Also, you can make the package look like it wraps an arbitrary present. –  Adam Zalcman Jan 2 '12 at 13:19
One consequence of this is that inertia and visible shape may be unrelated: think of an object made of two materials: one very dense and one very light. The dense core can take a small and complex shape that determines the mass, CoM and tensor of inertia. The light envelope defines externally visible shape. This is taken advantage of in some toys that automagically maintain balance. –  Adam Zalcman Jan 2 '12 at 13:26