october2013 47© 2013 The Royal Statistical Society

To o t s i e Po p sH o w m a n y l i c k s t o t h e c h o c o l a t e ?

Every American has grown up with the iconic Tootsie Pop commercial of the little boy trying to find the answer to the question “How many licks to the centre of a Tootsie Pop?”. For those from other cultures, here’s a quick recap.

Little Boy: “Mr. Owl, how many licks does it take to get to the Tootsie Roll centre of a Tootsie Pop?”

Mr. Owl: “Let’s find out. A one…two-hoo…a three…” (crunch sound effect as the temptation to bite becomes too great).

(Narrator): “How many licks does it take to get to the centre of a Tootsie Pop? The world may never know.”

The world was left in a haze of unknowing; as with Fermat’s last theorem, it was a question that begged to be answered. The Tootsie Pop problem was not quite as old as Fermat’s (1637 for that one), but the question did start back in 1931 when the Tootsie Pop itself was invented. The question was formalized in 1970 when the iconic Tootsie Pop commercial first aired. For those who have never seen it, of for those who need a nostalgia fix, it is at www.youtube.com/watch?v=K2xMGI-QpZw.

In 2008, Tootsie Pop tried to capture the public’s eye by hosting a contest to try to find the number of licks to the centre. The rigorous question has been attempted before by various universities, using various techniques ranging from student lick-ins to mechanical licking ma-chines. The average number of licks ranged from

Swarthmore College’s 144 licks (obviously big tongues) to as many as Cambridge’s 3481 licks. Clearly there was considerable variability here, and the question was never answered definitively. Even more important, there has been little or no documentation as to the experimental methods used for determining these historical results. It appears that none of the experiments were published, except, of course, on the somewhat less-than-reliable Internet.

I decided to find the number of licks by doing some laboratory experiments that focused on key variables that might be considered important to reaching the centre. My rationale was that lab experiments are cheaper than bribing students to lick, and that I was not at a university wealthy enough to finance me to build a licking machine of sorts. But necessity is the mother of inven-tion: I was able to break down my licking experi-ments, isolating three different variables that I considered would have the greatest effect on the number of licks. My variables were: the force of a lick, the temperature of a person’s tongue, and the pH of a person’s saliva. I simulated these ef-fects in a lab by doing a force test that involved a stir plate and some water.

Placing a Tootsie Pop in beaker of water (us-ing 150 ml of water for each test) I was able to change the speed of the stir plate to simulate an increased amount of force. I did several tests at four different speeds and used Minitab on the data. ANOVA and a Tukey post hoc test indicated that force of lick was not likely to affect the number of licks to reach the centre of a normally shaped Tootsie Pop.

Repeating the experiment with hotter and cooler water temperatures indicated no differ-ence as long as the temperatures were near that

of the human body The significant difference was observed at 97°C; well above – indeed danger-ously above – normal human licking capability. I concluded that for human lickers, their body temperature would not affect the number of licks.

My last test was to determine if pH levels and solubility of a person’s saliva might be a deter-mining lick factor. After doing four independent tests that considered normal pH, slightly basic pH, slightly acidic pH and a solubility test I was able to do a final ANOVA and come back with p-value of 0.334 – clearly not significant. This suggested that a person’s pH or the amount of saliva they secrete does not affect the number of licks.

I was now able to persuade several mathemat-ics classes to lick some Tootsie Pops for me. My preliminary results meant that I did not have to worry about how acidic a person’s saliva was, nor what their body temperature was, nor how hard they pressed on the lollipop with their tongue. Thus armed, I was confident I had something that would help me find the number of licks to the centre. For consistency, I asked the students to lick on the non-banded side of the Tootsie Pop – the sweet has a thicker ridge that runs longitudinally from its North to its South poles – and to lick only on one side. They could proclaim they had found the centre after they were able to “taste chocolate”. After entering the data, I was able to come up with the following results:

• Volunteers, 92• Mean licks, 356.1• Lick range, 78–1087• Inter-quartile range, 219.3–479.8• Standard deviation, 185.7

Tootsie Pops are lollipops on sticks with a ball of chocolate inside them. How many licks does it take to reach the chocolate? Every American child has wondered. Cory Heid set out to find the answer.

Table 1. Previous studies. Number of licks to the centre of a Tootsie Pop

Licking machines Licking experiments

Purdue University – 364 Swarthmore College – 144University of Michigan – 411 University of Cambridge – 3481Harvard University – 2255 Purdue University – 252

Source: http://en.wikipedia.org/wiki/Tootsie_PopsCredit: Gilabrand at en.wikipedia.org

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My mean number of licks came back within the range of the other universities’ results, which was encouraging; however, recall that that range was huge. My new concern was about the standard deviation, which was 186 licks – more than half the size of the mean. Yikes! The mas-sive variability of the licks was too great for my liking and I immediately started to ponder ways to explain it or to produce more consistent results. Were some of my initial assumptions incorrect?

After much deliberation I figured the best way to find out was to “cut to the core” of the Tootsie Pop itself. A heated sharp knife can slice through the lollipop with as much ease as cut-ting warm butter. For consistency I cut each one along the stick and along the longitudinal thick band. The results were quite interesting, as the photograph in Figure 1 shows…

I found that the supposed “centre” was not actually at the centre at all. Nor was the Tootsie Pop centre even spherically shaped as I had

assumed. In fact, no two centres were alike. I gathered more data on the Tootsie Pops’ height, width, flavour, core height, core width, shell thickness on the right-hand and left-hand sides and stick length (in millimetres) that was con-tained within the lollipop. Many of the different factors that I looked at varied greatly. The centre width (mean of 13 mm), centre height (mean of 21 mm) and stick length (18 mm) would vary as much as 3 mm between lollipops (see Table 2).

Each had a large amount of variation. Even the length of the stick within the sweet varied hugely, although this was something I assumed would be consistent. But its range was extreme, from 9.97 mm to 23.17 mm. Probably the most interesting part of all these open Tootsie Pops was the different shapes of their core. The exposed lollipops did not even have a circular two-dimensional shape; most of them were just irregular blobs of chocolate. Clearly we can see (Figure 2) that the centre is not circular (or spherical) and not very consistent.

While the centre of the Tootsie Pop was not what I had expected, some luck happened to find me. By accidently cutting one Tootsie Pop through the longitudinal band (and still cutting along the stick), I noticed that the core seemed to much thinner than in all the other lollipops that I had sliced open. This got me thinking that maybe the Tootsie Pop core is flatter, and more “coin”-shaped than spherical. If so, would I get better results by licking from a side where the centre was at a more consistent distance from the outside? So I started to re-employ my old methods of commissioning human lickers, but this time having them lick on the thick longitudinal band of the Tootsie Pop in-stead of the non-banded part. After a small pilot study of seven individuals at the Joint Statistical Meeting 2013 in Montreal I was able to collect the data shown in Table 3.

Eureka! By simply turning the Tootsie Pop to its side and licking along the band, I was able to find the centre with much smaller variability. Sure, it took on average 60 more licks (417 licks) to do so, but I was able to reduce the variation down to a mere 39 licks! My previous standard deviation, remember, was a massive 185; we have reduced that by 146 licks. By standardis-ing the side of the Tootsie Pop which was licked, we had at last achieved a reasonable degree of consistency. I can now more accurately predict the number of licks it takes to the centre. Since I had such a small sample size, I now turn to you, the reader and ask for your help to confirm or deny my results. Please lick your Tootsie Pop on the thick longitudinal band, and send your data to me at fatkenyan23@hotmail.com.

While Tootsie Pop may stick by their long lasting slogan, “The world may never know”, at least I can say “You may not know, but I do. It is 417 – give or take 39 licks.”

Cory Heid is at Siena Heights University, Michigan. He is interested in data analysis projects and data-driven model building.

Figure 1. Tootsie Pops cut longitudinally

Table 2. Some vital statistics of the Tootsie Pop. ‘Shell R’ = shell thickness on right-hand side, ‘Shell L ‘ = shell thickness on left-hand side. Dimensions in mm. Red circles indicate extreme variability of the length of the stick within the sweet

Variable Count Mean StDev Minimum Q1 Median Q3 Maximum

Height 12 31.648 1.096 30.440 30.665 31.435 32.118 33.930Width 12 29.015 1.448 26.010 28.232 28.965 29.655 31.610Core height 12 21.428 2.483 16.890 19.712 21.380 23.667 25.260Core width 12 13.207 2.694 8.720 10.578 13.775 15.693 16.560Shell R 12 5.870 1.597 3.420 5.005 5.620 6.475 9.450Shell L 12 6.991 1.598 4.820 6.183 6.900 7.635 11.060Stick length 12 18.38 3.51 9.79 16.89 18.81 20.87 23.17

Table 3. Descriptive statistics: licks

Variable N Mean SE mean StDev Minimum Q1 Median Q3 Maximum

Licks 7 417.9 14.9 39.3 380.0 396.0 409.0 430.0 500.0

Figure 2. Asymmetrical and non-spherical centres of the Tootsie Pop