Anna and Boris play a game with a red stick, a white stick and a blue stick, each of which is 1 meter long. Anna starts by breaking the red stick into three pieces. Then Boris breaks the white stick into three pieces. Finally, Anna breaks the blue stick into three pieces. She wins if she can use the nine pieces to form three triangles with sides of different colors. Can Boris stop her from winning?

Clarification: You can assume the sticks are straight line segments and when they are broken, the fragments are straight line segments whose lengths sum to 1 meter.

2nd clarification: For each triangle that Anna makes, there must be a red side, a white side and a blue side.

This puzzle is from a Leningrad Mathematical Olympiad.


1 Answer 1


Boris cannot stop Anna from winning.

In fact

Anna can break both her sticks before Boris breaks his and still win.

Anna breaks both of hers as $1/2, 1/4, 1/4$ She can match the two long sticks with the longest of Boris'. As it must be shorter than $1$ she can form a triangle. Two $1/4$ sticks can make a triangle with any segment of length in $(0,1/2)$. Both of Boris' smaller sticks still must have positive length and cannot be $1/2$ or more in length.

  • 1
    $\begingroup$ In general, Anna can rot13(oernx obgu bs ure fgvpxf vqragvpnyyl jvgu yratguf ng yrnfg bar unys, bar sbhegu, naq bar fvkgu.) $\endgroup$
    – isaacg
    Mar 8 at 4:06
  • 2
    $\begingroup$ @isaacg: that is correct. V fhfcrpg gurer ner jnlf jurer gur fgvpxf ner abg oebxra gur fnzr nf jryy naq pregnvayl jnlf hfvat gur serrqbz gb ernpg gb Obevf' oernxvat. $\endgroup$ Mar 8 at 4:30
  • $\begingroup$ @isaacg rot13(bar, guerr naq fvk graguf jvyy nyfb jbex (vs gur fubegrfg bs Obevf vf bire gjb graguf gur 1 naq 3 arrq gb or pbzovarq)) $\endgroup$
    – Retudin
    Mar 8 at 17:52

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