Chase Me, Catch Me

Chase Me, Catch Me $Description: C:\G Drive\niquette\images\solution.gif$

$Description: C:\G Drive\niquette\pictures\cmcmfig1.gif$ $Description: C:\G Drive\niquette\images\flam-l.gif$ et us begin our analysis of the puzzle with Figure 1, which shows a schematic representation of a person walking in low-heel shoes, superimposed upon which is a person walking in high-heel shoes. The shapes and dimensions of the foot are derived from the x-ray photograph in the Chase Me, Catch Me illustration. In the natural stride, the heel of the foot at its back-most position is elevated and thus not acted upon by the high heel. In its forward-most position, the heel of either foot (whatever its shodding) must engage the walking surface. The high-heel dimension has been chosen to assure that the angle at the ankle is at its limit, which keeps the shin nearly vertical and the forward knee necessarily bent. As indicated, the stride-angle at the hip is held constant between the two walkers.

A direct comparison in stride length in Figure 1, along with simple geometrical relationships, will provide an immediate solution to the puzzle: The wearer of high-heel shoes experiences the shortening of each stride by...

(a) 1/4th

That, presumably, makes the chase unfair, the catch easy. If you are curious about the three remaining choices, you will find that they are all correct -- or can be correct, depending on a few assumptions (see Vetruvian Man).

$Description: C:\G Drive\niquette\images\flam-w.gif$ alking with only two feet is a precarious activity, if you stop to think about it, which few of us ever do. Walking is really what might be called "arrested falling." Notice that Figure 1 postulated the location of the walker's center of gravity (CG), generally at about the level of the navel. The CG serves a vital function in upright locomotion, for we observe that the CG must always be positioned some distance ahead of the back-most location of the walker's foot. The incipient fall is arrested by the action of the other foot, of course, when it strikes the surface in its forward-most position.

$Description: C:\G Drive\niquette\pictures\cmcmfig2.gif$ Perhaps you noticed in Figure 1 above that for the low-heel walker, the CG was assumed to be located directly above the hips. For the walker in high-heels, the hips will be drawn farther aft, closer to the back-most location of the walker's foot. That shortens the "moment arm" (leverage) for empowering forward motion, resulting in a slower walking speed. The remedy is elementary. The wearer of high-heeled shoes merely leans forward at the hip to move the CG to about the same horizontal location as that enjoyed by the wearer of low-heeled shoes. And the chase goes on.

Aviation terminology has been appropriated here to describe maneuvers of the pelvis as it flies gracefully along above the surface. Thus "pitch" appeared in Figure 1 for horizontally positioning the CG. Meanwhile, "roll" and "yaw" were both postulated as zero for that case.

$Description: C:\G Drive\niquette\images\flam-f.gif$ igure 2 addresses another aspect of the walking challenge, keeping the body moving along at a constant elevation above the surface. The foot wearing a high-heeled shoe passing under the body during the stride will make lifting of the CG decidedly more pronounced. Pelvic roll can be applied to reduce the amount of vertical motion by lowering the hip connected to the forward-most leg. That does increase the flexing at the knee, however, which diverts some amount of the walker's effort, making the chase even more unfair.

$Description: C:\G Drive\niquette\pictures\cmcmfig3.gif$ All aesthetic considerations have been set aside in this analysis. The Chase Me, Catch Me puzzle is interested only in relative stride-length, to which we now return...

Shown in Figure 3 is the high-heel walker now deploying plenty of pelvic yaw, with the apparent intention of lengthening the stride -- recovering some of the disadvantage in the chase. The note in Figure 3 is salient, however, for we are reminded that the walker postulated for the low-heel reference stride does not make use of pelvic stratagems. Indeed, all proposed improvements in the high-heel stride must be offset by the fact that for this simplified (ceteris parabus) comparison, the low-heeled walker is unfairly constrained.

Ah, but there is not a dress-code on the planet that forbids the wearer of low-heeled shoes to go pitching, rolling, and yawing along the sidewalk enroute to workplace (backpacking a pair of high-heeled shoes, presumably).

Background

First heard in a high school locker room back in the forties, the beckoning title of the Chase Me, Catch Me puzzle is a shortened version of enticements wishfully attributed to young ladies in high-heels. The complete version appended the phrase “wed me, bed me" (in that order, of course).

$Description: C:\G Drive\niquette\pictures\vetman.jpg$ The Vetruvian Man (The Man in Action) sometimes called the "Canon of Proportions" is a drawing with accompanying notes by Leonardo da Vinci made around the year 1490 depicting a human figure in two superimposed positions with arms apart and simultaneously inscribed in a circle and square. That drawing is at least a place to start. Relative to height, we find thigh = shin = 26.1%; foot = 12.4%; ankle-to-ball-of-foot = 9.1%.

Taking a range of heights for most likely wearers of high-heel shoes as 5' 4" to 5' 9", a median heel-height of 4", and a median low-heeled stride-length of 2' 6", we see the equivalance of....

(a) 1/4th reduction in length of stride,
(b) 7¹/2 inches,
(c) 1⁷/8 times the heel-height, and
(d) 1¹/2 times the foot-print (the expression "foot-print" splits the difference between that produced by high- and low-heel).

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The Puzzle as a Literary Genre

Epilog

As the author of Chase Me, Catch Me in 2005, I offered a technical analysis of merely one aspect in foot fashions.

Comme auteur de Chase Me, Catch Me en 2005, j'ai offert une analyse technique d'un seul aspect des modes du pied.

As a subscriber to the New York Times, I have studied The Daily Shoe and its imperatives for that same aspect in foot fashions.

Comme abonné au New York Times, j'ai étudié The Daily Shoe et ses impératifs pour ce même aspect des modes du pied.

As an American living in France since 2012, I have observed the discomfort imposed by that same aspect in foot fashions.

Comme un américain vivant en France depuis 2012, j’ai observé la douleur imposée par ce même aspect des modes du pied.

As a sympathetic man, I now understand why the most fashionable women in France walk like farm animals.

Comme homme sympathique, je comprends maintenant pourquoi les femmes les plus à la mode marchent comme des animaux de ferme.