<p>If: </p> <ul> <li>a<sup>2</sup> – bc = 5 </li><li> b<sup>2</sup> – ac = 7</li><li> c<sup>2</sup> – ab = 9</li></ul> <p>Then what are a, b, and c? </p>

<div><button onclick='var btn = this; var div = btn.parentNode.childNodes[1]; if (div.style.display != "block") { div.style.display="block"; } else { div.style.display="none"; }' style='padding:2px 4px;'>Answer</button><div class='spoiler'> <p>Solution 1: </p> <ul> <li>a = -sqrt(1805/315) =~ -2.393775 </li> <li>b = +(2/19)×sqrt(1805/315) =~ 0.251976 </li> <li>c = +(23/19)×sqrt(1805/315) =~ 2.897728 </li> </ul> <p>Solution 2: </p> <ul> <li>a = +sqrt(1805/315) =~ +2.393775 </li> <li>b = -(2/19)×sqrt(1805/315) =~ -0.251976 </li> <li>c = -(23/19)×sqrt(1805/315) =~ -2.897728 </li> </ul> </div></div> <p>Here is my <a href="/wizfiles/pdf/abc-puzzle.pdf">solution</a> (PDF). </p>

<p>What is the expected number of spins in roulette to see five reds or five blacks in a row? </p>

<div><button onclick='var btn = this; var div = btn.parentNode.childNodes[1]; if (btn.innerHTML == "Show Spoiler") { div.style.display="block"; btn.innerHTML = "Hide Spoiler"; } else { div.style.display="none"; btn.innerHTML = "Show Spoiler"; }' style='padding:2px 4px;'>Show Spoiler</button><div class='spoiler'> The answer is 3872789/118098 =~ 32.79301089 spins. </div></div> <p> </p> <p>Here is my <a href="/wizfiles/pdf/roulette-five-same-color-ver2.pdf">solution</a> (PDF). </p>

<p>Upon opening a Facebook account, the following are the average number of ads for fake stamps received by number of days since opening the account: </p> <ul> <li>4 ads within one day. </li> <li>16 ads within two days. </li> <li>36 ads within three days. </li></ul> <p>What is the expected number of hours until the first ad for fake stamps appears? </p>

<div><button onclick='var btn = this; var div = btn.parentNode.childNodes[1]; if (div.style.display != "block") { div.style.display="block"; } else { div.style.display="none"; }' style='padding:2px 4px;'>Answer</button><div class='spoiler'> The answer is sqrt(π)/4 days =~ 10 hours, 38 minutes, 5 seconds. </div></div> <p> </p> <p>Here is my <a href="/wizfiles/pdf/fake-stamps.pdf">solution</a> (PDF). </p> <p>This question is asked and discussed in my forum at <a href="https://wizardofvegas.com/forum/questions-and-answers/math/34502-easy-math-puzzles/78#post892891" target=_blank>Wizard of Vegas</a>. </p>

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Ask The Wizard #384

Ask The Wizard #384

If:

a²– bc = 5
b²– ac = 7
c²– ab = 9

Then what are a, b, and c?

anonymous

Solution 1:

a = -sqrt(1805/315) = -2.393775
b = +(2/19)×sqrt(1805/315) = 0.251976
c = +(23/19)×sqrt(1805/315) = 2.897728

Solution 2:

a = +sqrt(1805/315) = +2.393775
b = -(2/19)×sqrt(1805/315) = -0.251976
c = -(23/19)×sqrt(1805/315) = -2.897728

Here is my solution (PDF).

How many rounds of roulette are statistically required to witness five consecutive red or black outcomes?

anonymous

The answer is 3872789/118098 = 32.79301089 spins.

Here is my solution (PDF).

When you create a Facebook account, you can expect to see a certain number of advertisements for counterfeit stamps over time. calculated based on the number of days since your account was activated: