Blank Canvas Arts and Crafts

Paper Punches

A Paper Punch Problem?

Consider a paper punch that can be centered at any point of the plane, and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point?

I think the idea of the question is that every point which is ANY irrational distance from the center is removed (a rather odd looking hole-punch). Then we want to recenter and apply again, etc.

If we stick with standard coordinates, we may as well assume that our first punch is centered at (0,0). Then of course we don’t actually remove (0,0), so let’s take our second center to be, say, (pi, 0) [now here there might be a difference in what point we pick...]. Then we’ve now removed every point of the x-axis: rationals are removed since their distance to pi is irrational; irrationals were removed since the distance to 0 is irrational.

Then the question is which other points of the plane are left out, and how many extra punches we need. [And perhaps we ought to have chosen a different second point, perhaps an algebraic, or maybe off of the x-axis.]

If we assume that (a,b) in R^2 such that sqrt(a^2+b^2) and sqrt((a-pi)^2+b^2) are rational, then we can find that a=pi/2, b=sqrt(r^2-(pi/2)^2) work out for almost any rational r. So we are missing at least some points, along the vertical line x=pi/2.

Mer, late for work…

EDIT: slow day at work
I think perhaps it would be best to start with an algebraic point, say A=(sqrt(2), 0). If we punch there and at O=(0,0), assume that there is a point (a,b) such that its distance to O and A is rational, i.e.
sqrt(a^2+b^2) in Q
sqrt(a^2 + b^2 + 2 – 2a*sqrt(2)) in Q
Let r^2=a^2+b^2 (which is rational). Then we need
r^2 + 2 – 2a*sqrt(2)
to be rational (indeed, it must be the square of a rational); this implies that a=s*sqrt(2) for some rational s.

So there may be some points which we have missed, but if they do exist they all are on the family of vertical lines x=s*sqrt(2). Now we can take the point B=(sqrt(3), 0), and if there is a point (a,b) with rational distance from B and O, we similarly obtain that
r^2 + 3 – 2sqrt(3)*a
is rational. If it further has rational distance to A, then we know a=s*sqrt(2), so
r^2 + 3 – 2s*sqrt(6)
is rational, which says that sqrt(6) is rational, which is nonsense.

Thus [unless I've missed something], three punches can suffice. I strongly suspect two cannot do the job; I think that given two centers, there will be points on the perpendicular bisector of the two which were not removed.

EDIT: I’m certain that two cannot do the job. For any two points in the plane, pick some rational number larger than half the distance between them, and construct the isosceles triangle(s) with the segment joining our two points as the base and equal legs of length that chosen rational.

One begins to wonder about higher dimensions, and perhaps even different metrics on the space…

EDIT: holdm: “then sqrt(x^2+y&2) is rational
in this case, sqrt((x-e)^2 + y^2) is irrational”
Consider (x,y)=(e/2, sqrt(25-(e/2)^2)). Then
sqrt(x^2+y^2) = 5 is rational, but
sqrt((x-e)^2 + y^2) = 5 is also rational.

Also, I think Gerry was thinking of just the real line instead of the plane. He raises the interesting question though whether we can center a punch at an already-removed point. If we can’t, we’ll need infinitely many punches. Suppose instead that there is a last punch. Then the last punch is centered at some point X which has not yet been removed. But the last punch won’t remove X, being rational distance away from itself. An interesting question then is to have our punch also remove the point on which it is centered. I’m not sure how to go about choosing our points then…

EDIT (after Vasek’s answer): The higher dimensional question seems to be actually rather uninteresting. The same three punches will suffice for any dimensional space (where any new coordinates are taken to be zero for our punches). The argument is the same as above, and the only time we can get away with fewer than three punches is when we’re dealing with just the real line. (And I suppose you can’t remove everything in R^0, if you take that to be a single point…)

Paper Punch Storage System

Pattern Craft Punch Shape Smile Happy for paper card M $6.49

Pattern Craft Punch Shape Smile Happy for paper card L $8.90

Small Fuchsia Gray 2 Holes Paper Metal Punch Stationery $7.11

FLOWERS LEAVES & MORE l@@k @ photo examples paper punch bunch SUPER GIANT $21.99

18Pcs Paper Punches Crafts Scrapbook Edge Decoration Assorted $52.48

Small Balloon Paper Punch Quilling-Scrapbook-Cardcraft $5.40

TAG EK Success Paper Shapers Punch Large Slim $13.49

EK Success SLIM EDGER PAPER PUNCH ~ SNOWFLAKE $13.99

1″ ROUNDER Slim CORNER Paper Shapers Punch EK Success $8.49

Single Round Hole Craft Paper Punch NEW $4.49

Cutting Edge Paper Punch Set $15.00

Creative Memories Paper punch holder Cary Case Pre-cut Star Shaped $2.99

Creative Memories Paper punch holder Cary Case Pre-cut Circle Shaped $2.99

Creative Memories Paper punch holder Cary Case Pre-cut Circle Shaped Red $2.99

CARD MAKING KIT,JOURNALING PAPER SHAPES, PUNCH-OUTS $5.87

Wilton 570-1121 Easy Flex 3-Piece Silicone Spatula Set, Blue $5.91 Silicone Baking Accessories. Silicone is the Wonder Kitchen Tool: it can be used in the oven (up to 500F/260C), toaster over, microwave, refrigerator and freezer and it is virtually unbreakable. These silicone utensils are the perfect tool cold foods: Jell-O, ice sculptures, Candies and hot foods: cake, muffins, bread, pies, main and side dishes and so much more! The flexibility makes removing foo…

Wilton Decorating Basics Lesson Plan $2.30 WILTON-Decorating Basics Lesson Plan. This book is a great introduction to creating beautifully decorated cakes at home! This lesson plan is made to coordinate with the Wilton Decorating Basics Student Kit (not included) and is full of great ideas for the beginner cake decorator. This book features basic decorating principles; preparation; and detailed instructions for constructing stars; drop flo…

Wilton Flowers and Cake Design Lesson Plan $2.30 Flowers and Cake Design Lesso Plan. This book contains useful printed tips and techniques for anyone looking to expand their knowledge of cake design and floral cake decorating. This lesson plan is made to coordinate with the Wilton Flowers and Cake Design Student Kit (not included) and is full of great ideas whether you are a beginner or advanced cake decorator. This book features basic cake desi…

Piano Music for Children $19.98 …

A Few Useful Tips About Living Underground $6.30 ranger perfect pearls brushes. now you don t have to worry about getting your pigments mixed up when you don t want them to be. this package contains twelve detail brushes just like the one found in the perfect pearls pigment kits sold separately. imported….

Let’s Party $3.00 AMSCAN-Let’s Party CD. Bring the rhythm to you party with Drew’s Famous Let’s Party. Songs include I Got You (I Feel Good); Super Freak; American Pie; Money Money and more. This package contains one cd with 18 pub songs. Made in USA….

Bob Ross: Getting started with Oil Colors [VHS] $6.21 Getting Started with Bob Ross DVD. An introduction to Bob Ross’ popular Wet-on-Wet painting technique. Bob Ross, host of TV’s favorite ‘Joy of Painting’ show, gives you an overview of his technique and shows you how to paint ‘Mountain Summit’ using his unique materials (materials sold separately). This full-screen DVD features English voice and subtitles plus Spanish, French, Mandarin Chinese, Ger…

SE Leather Hole Punch Tool, Heavy Duty 2.0mm – 4.5mm $1.18 9 1/4 in. Professional Leather Punch Plier with Revolving Head and Replaceable Punch Sizes 2.0mm, 2.5mm, 3.0mm, 3.5mm, 4.0mm, 4.5mm…

TRENDnet RJ-11/RJ-45 Crimp/Cut/Strip Tool TC-CT68 (Blue) $13.10 Our professional crimp tool TC-CT68 crimps, strips, and cuts cable for 8 and 6 position modular plugs. The crimp tool offers reliable, easy-to-use, and affordable solution to meet immediate cutting needs. Our professional crimp tool delivers convenience at a very economical price….

TEKTON 6610 5/32-Inch Letter and Number Stamp Set, 36-Piece $19.39 MIT LETTER & NUMBER PUNCH features: * Heat Treated, Chrome Vanadium Steel Punches * Black-Oxide Finished for Durability * 36 Punches Marked A-Z and 0-9 * Stamp Metal, Wood, Plastic and Leather * Permanently Imprint Names, Initials, and Codes on Personal Belongings * Helps Deter Theft and Recover Stolen Items * Punches Store in Stand-Up Wooden Index Box * Size: Shaft Length 2-3/8″, Shaft Width 1/4″…